Extremal problems in uniformly dense hypergraphs
Christian Reiher
TL;DR
This paper explores extremal problems in uniformly dense hypergraphs, focusing on maximum edge counts avoiding certain subhypergraphs, and discusses recent results and open problems using hypergraph regularity methods.
Contribution
It introduces new extremal problems for uniformly dense hypergraphs and applies hypergraph regularity techniques to obtain recent results and identify open questions.
Findings
Hypergraph regularity method is effective for extremal problems.
Results on maximum edges in hypergraphs avoiding specific subhypergraphs.
Open problems in uniformly dense hypergraph extremal theory.
Abstract
For a -uniform hypergraph let be the maximum number of edges of a -uniform -vertex hypergraph which contains no copy of . Determining or estimating is a classical and central problem in extremal combinatorics. While for graphs () this problem is well understood, due to the work of Mantel, Tur\'an, Erd\H{o}s, Stone, Simonovits and many others, only very little is known for -uniform hypergraphs for . Already the case when is a -uniform hypergraph with three edges on vertices is still wide open even for . We consider variants of such problems where the large hypergraph enjoys additional hereditary density conditions. Questions of this type were suggested by Erd\H{o}s and S\'os about 30 years ago. In recent work with R\"odl and Schacht it turned out that the regularity method for hypergraphs,âŚ
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Extremal problems in uniformly dense hypergraphs
Christian Reiher
Fachbereich Mathematik, Universität Hamburg, Hamburg, Germany
Abstract.
For a -uniform hypergraph let be the maximum number of edges of a -uniform -vertex hypergraph which contains no copy of . Determining or estimating is a classical and central problem in extremal combinatorics. While for graphs () this problem is well understood, due to the work of Mantel, TurĂĄn, ErdĹs, Stone, Simonovits and many others, only very little is known for -uniform hypergraphs for . Already the case when is a -uniform hypergraph with three edges on vertices is still wide open even for .
We consider variants of such problems where the large hypergraph enjoys additional hereditary density conditions. Questions of this type were suggested by ErdĹs and SĂłs about 30 years ago. In recent work with RĂśdl and Schacht it turned out that the regularity method for hypergraphs, established by Gowers and by RĂśdl et al. about a decade ago, is a suitable tool for extremal problems of this type and we shall discuss some of those recent results and some interesting open problems in this area.
Key words and phrases:
TurĂĄnâs hypergraph problem, uniformly dense hypergraphs, hypergraph regularity method
2010 Mathematics Subject Classification:
05C35 (primary), 05C65, 05C80 (secondary)
1. Introduction
1.1. TurĂĄnâs extremal problem
Extremal graph theory is known to have been initiated by TurĂĄnâs seminal article [Tu41], in which he proved that for there is, among all graphs on vertices not containing a clique of order , exactly one whose number of edges is maximal, namely the balanced complete -partite graph. TurĂĄn then asked for similar results, where instead of a clique one intends to find the -skeleton of a given platonic solid in the host graph. Moreover, he proposed to study analogous questions in the context of hypergraphs.
Fixing some terminology, we say for a nonnegative integer that a pair is a -uniform hypergraph, if is a finite set of vertices and is a set of -element subsets of , whose members are called the edges of . As usual -uniform hypergraphs are simply called graphs. Associated with every given -uniform hypergraph one has TurĂĄnâs extremal function mapping every positive integer to
[TABLE]
i.e., to the largest number of edges that a -uniform hypergraph on vertices without containing as a (not necessarily induced) subhypergraph can have. In its strictest sense, TurĂĄnâs hypergraph problem asks to determine this function for every hypergraph .
Using an averaging argument, Katona, Nemetz, and Simonovits [KNS64] have shown that for every -uniform hypergraph the sequence n\longmapsto\operatorname{ex}(n,F)\big{/}\binom{n}{k} is nonincreasing. Therefore the limit
[TABLE]
known as the TurĂĄn density of , exists. The problem to determine the TurĂĄn densities of all hypergraphs is likewise called TurĂĄnâs hypergraph problem.
It may be observed that these problems are trivial for , while the case is fairly well understood. TurĂĄn himself [Tu41] determined for all integers and , thus proving for every integer . This was further generalised by ErdĹs and Stone [ErSt46], and their result can be shown to yield the full answer to the TurĂĄn density problem in the case of graphs. Explicitly, we have
[TABLE]
for every graph with at least one edge, where denotes the chromatic number of , i.e., the least integer for which there exists a graph homomorphism from to (see also [ErSi66], where the connection with the chromatic number appeared first).
Already for , however, our knowledge is very limited and there are only very few -uniform hypergraphs for which the function is completely known. A notable example occurs when denotes the Fano plane. SĂłs conjectured in the 1970s that for the balanced, complete, bipartite hypergraph is extremal for this problem. The first result in this direction is due to de Caen and FĂźredi [DeFu00], who proved that at least the consequence of SĂłsâs conjecture holds. By combining their work with Simonovitsâs stability method [Si68] it was shown in [FuSi05, KeSu05] that the conjecture holds for all sufficiently large hypergraphs. A full proof applying to all was recently obtained in [BR].
On the other hand, even concerning the -uniform hypergraphs on four vertices with three and four edges, denoted by and respectively, it is only known that
[TABLE]
The lower bounds are derived from explicit constructions due to Frankl and FĂźredi [FrFu84] and to TurĂĄn (see, e.g., [Er77]), and in both cases they are universally believed to be optimal. The upper bounds were obtained by computer assisted calculations based on Razborovâs flag algebra method introduced in [Ra07]. They are due to Baber and Talbot [BaTa11], and to Razborov himself [Ra10]. For a more detailed discussion of our current knowledge about Turanâs hypergraph problem we refer to Keevashâs survey [Ke11].
1.2. Uniformly dense hypergraphs
Let us now restrict our attention to -uniform hypergraphs. Accordingly, the word hypergraph will henceforth always mean -uniform hypergraph. Concerning the extremal problem for it was thought for a while that its TurĂĄn density might be .
This notion was based on the following construction, which goes back to the work of ErdĹs and Hajnal [ErHa72]. Take a random tournament on a large set of vertices. Evidently any three vertices in induce either a transitive subtournament of or a cyclic triangle. Furthermore, the former happens with a probability of and the latter with a probability of . Define, depending on , a random hypergraph on whose edges correspond to the cyclic triangles in . One checks easily that can never contain a and the random choice of causes to have, with high probability, an edge density close to .
While the construction of Frankl and FĂźredi [FrFu84] mentioned earlier shows that the hypergraphs cannot be optimal among all -free hypergraphs, it was suggested by ErdĹs and SĂłs (see e.g., [ErSo82, Er90]) that there might still be a natural sense in which they are optimal -free hypergraphs. Specifically, they suggested to focus only on uniformly dense host hypergraphs defined as follows.
Definition 1.1**.**
For real numbers and we say that a -uniform hypergraph is uniformly -dense if for all the estimate
[TABLE]
holds.
Using standard probabilistic estimates one checks easily that for every accuracy parameter the probability that is uniformly -dense tends to as the number of vertices tends to infinity. The Turån theoretic question about the optimal density of uniformly dense hypergraphs not containing a given hypergraph (such as ) can be made precise by introducing the quantities
[TABLE]
which are to be regarded as modified versions of the usual TurĂĄn densities for uniformly dense hypergraphs. With this notation at hand, the tournament construction shows that \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(K_{4}^{(3)-})\geq\tfrac{1}{4} and the aforementioned conjecture of ErdĹs and SĂłs states that, actually, this holds with equality. Recently this has been shown independently in [GKV] and in [RRS-a].
Theorem 1.2**.**
We have \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(K_{4}^{(3)-})=\tfrac{1}{4}.
One of the two proofs referred to above consists of a computer-generated argument based on Razborovâs flag algebra method, while the other one uses the hypergraph regularity method. The subsequent progress in this area (see [RRS-e, RRS-zero]) has followed the latter approach. Moreover, continuing the collaboration with RĂśdl and Schacht, we have shown that there is a large number of further variants of the classical TurĂĄn density that can likewise be studied by means of the hypergraph regularity method (see [RRS-c, RRS-d]). The goal of this article is to survey these recent developments.
Before we proceed any further, however, we would like to draw the readerâs attention to perhaps the most urgent problem in the area, the determination of \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(K^{(3)}_{4}). The following construction, due to RĂśdl [Ro86], shows that this number has to be at least . Consider, for a sufficiently large natural number , the elements of as vertices. Assign to every pair of vertices uniformly at random one of the colours red or green. Declare a triple with to be an edge of the hypergraph we are about to exhibit, if the colours of and disagree. Of course this happens with a probability of and, again, standard probabilistic arguments show that for every it happens asymptotically almost surely that is uniformly -dense. Moreover, it is impossible that contains a tetrahedron. This is because for any four vertices it must be the case that two of the three pairs , , and receive the same colour, meaning that the three triples , , and cannot be present in at the same time.
Conjecture 1.3**.**
RĂśdlâs construction is optimal, i.e., we have \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(K_{4}^{(3)})=\tfrac{1}{2}.
A partial result in this direction is given by Theorem 1.4 below.
1.3. Further TurĂĄn densities
For proving results about \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(\cdot) one typically works with a property of hypergraphs that turns out to be more useful than the uniform density condition introduced in Definition 1.1. Rather than knowing something about the edge densities within single sets of vertices, it is more helpful to have comparable knowledge about the edge densities between any three sets of vertices. Explicitly, if denotes a hypergraph and , we set
[TABLE]
Moreover, for two real numbers and we say that is (d,\eta,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense if
[TABLE]
holds for all . One checks immediately by setting that every (d,\eta,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense hypergraph is also uniformly -dense. In the converse direction one can only show that large uniformly dense hypergraphs contain linear sized subhypergraphs that are still dense in this new sense with almost the same density, and that this is enough for proving
[TABLE]
A proof of this equality can be found in [RRS-e]*Proposition 2.5, where one has to set and . Alternatively, the reader may prefer to regard (1.3) as the âofficial definitionâ of \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(\cdot) and treat (1.2) just like an additional piece of information that is not going to be used throughout the rest of this article. As a matter of fact, this may even be the more natural approach to this subject, and the three dots occurring in the symbol \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(\cdot) are intended to remind us of the three sets , , and mentioned in the definition of being (d,\eta,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense.
We proceed with a more restrictive property of hypergraphs shared by both the random tournament construction and by RĂśdlâs hypergraph introduced in the previous subsection. Given a hypergraph , a set , and a set of ordered pairs we set
[TABLE]
So for instance E_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(A,B\times C)=E_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(A,B,C) holds for all . Next, for two real numbers and we say that is (d,\eta,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense provided that
[TABLE]
holds for all and . Finally we define
[TABLE]
for every hypergraph . Since every (d,\eta,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense hypergraph is, in particular, also (d,\eta,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense, we have
[TABLE]
for every hypergraph . Let us remark at this point that due to the fact that RĂśdlâs hypergraph is (\frac{1}{2},\eta,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense we have \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(K_{4}^{(3)})\geq\frac{1}{2}. Thus the following result from [RRS-c] shows that a considerably weaker version of Conjecture 1.3 is true.
Theorem 1.4**.**
We have \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(K_{4}^{(3)})=\tfrac{1}{2}.
The process of replacing a pair of sets by a set of pairs may be repeated once more. For a hypergraph and two sets of ordered pairs of vertices one defines
[TABLE]
as well as
[TABLE]
Notice that for all and we have
[TABLE]
Next, we declare to be (d,\eta,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense for two real numbers and if
[TABLE]
holds for all . If this is the case, then is (d,\eta,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense as well. The generalised TurĂĄn densities appropriate for this concept are defined by
[TABLE]
for every hypergraph , and as before we may observe that
[TABLE]
The investigation of these quantities was initiated in [RRS-d], where the case that is a clique received particular attention. This led to the curious situation that while the value of \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(K^{(3)}_{5}) is still unknown, it has been be shown that \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(K^{(3)}_{11})=\tfrac{2}{3} holds (see Theorem 2.9). We would like to mention that \mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}-dense hypergraphs have recently also been studied by Aigner-Horev and Levy [ELAD] in the context of hypergraph Hamiltonicity problems.
It is natural to expect at this moment some definitions of sets like \mathcal{K}_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(P,Q,R) and E_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(P,Q,R) involving three sets of ordered pairs, but it can be shown that the corresponding generalised Turån densities \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F) vanish for all hypergraphs (see [KRS02]).
Still, there are some further variations on this theme. We refer to the concluding remarks in [RRS-c] for a complete enumeration of all uniform density notions in the context of -uniform hypergraphs***Strictly speaking, that article deals with quasirandomness notions instead of density notions, the difference being that in [RRS-c] there are also upper bounds imposed on the numbers |E_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(P,Q)|, etc. It seems, however, that the present version demanding only lower bounds on these numbers is more natural from the perspective of hypergraph Turån problems.. A more systematic account applying to -uniform hypergraphs for all has been given in [RRS-e]*Section 2. In this survey, however, we shall mainly focus on the most concrete cases \mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ 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\leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ 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}\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}, and \mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ 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}\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}.
2. Examples
All known lower bounds on quantities of the form with \star\in\{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}},\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}},\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}\} are derived from probabilistic constructions that can be viewed as appropriate modifications of RĂśdlâs hypergraph introduced at the end of Subsection 1.2. Basically, these constructions combine an ordering of the vertices, a colouring of the pairs of vertices, and certain rules telling us which colour patterns on triples of vertices are to be translated into edges of the envisioned hypergraph.
As a matter of fact, even the ErdĹs-Hajnal tournament hypergraph can be presented in this manner, even though prima facie it depends on an orientation rather than on a colouring of the pairs. Once its vertices receive an arbitrary ordering, however, there will be âforwardâ and âbackwardâ arcs between the vertices, and this state of affairs can alternatively be encoded by using two colours. Moreover, one can decide the presence or absence of an edge in the hypergraph as soon as one knows the three âcoloursâ received by the pairs , , and (as well as the ordering of ).
For all these reasons, we shall now describe an abstract framework for presenting such constructions. Given a nonempty finite set of colours we call a set a palette (over ). So the elements of palettes are ordered triples of colours, called colour patterns. Such a palette is said to be (d,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense for a real number if holds. Given a vertex set equipped with a linear ordering and a colouring we define a hypergraph by
[TABLE]
In practice, one usually takes for a sufficiently large integer and adopts the standard ordering on this set as . This causes no loss of generality in the sense that one still considers the same isomorphism types of hypergraphs as in the general case.
Now the important observation is that if the underlying palette is (d,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense for some real , and if the colouring gets chosen uniformly at random (among all possibilities), then for every the hypergraph is asymptotically almost surely (d,\eta,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense. Furthermore, given a hypergraph and a palette it can be decided in finite time whether there exists a hypergraph of the form containing . Specifically, this happens if and only if there exists an ordering of as well as a colouring of the set of pairs covered by edges of such that every edge of with satisfies
[TABLE]
Thus, whenever fails to admit such a pair , one knows that \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)\geq d.
Example 2.1**.**
The simplest (nontrivial) palettes that can be imagined just consist of a single colour pattern. Owing to potential repetitions of colours in such a pattern, there arise several distinct possibilities, the most restrictive of which is given by three distinct colours. So let us consider the case that and .
Clearly, is (\tfrac{1}{27},\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense. Therefore, the previous discussion shows that if a hypergraph does not have property (*â*) ⣠2.2 in Theorem 2.2 below, then \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)\geq\tfrac{1}{27}. In other words, if \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)<\tfrac{1}{27}, then needs to admit such an ordering of its vertices together with such a colouring of its shadow. The main result of [RRS-zero] informs us that under this condition one actually has \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)=0. This implies that \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)\not\in(0,\frac{1}{27}) holds for every hypergraph .
Theorem 2.2**.**
For a -uniform hypergraph , the following are equivalent:
- (*â*)
\pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)=0. 2. (*â*)
There is an enumeration of the vertex set and there is a three-colouring of the pairs of vertices covered by hyperedges of such that every hyperedge with satisfies
[TABLE]
Example 2.3**.**
As indicated by the discussion in the second paragraph of this section, the tournament hypergraph can be defined by the (\tfrac{1}{4},\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense palette
[TABLE]
over . The proof of Theorem 1.2 presented in [RRS-a] proceeds by showing that for every (\tfrac{1}{4}+\varepsilon,\eta,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense hypergraph on vertices possesses a vertex whose link graph contains a triangle. It thus seems natural to wonder whether similar ideas can be used to settle the value of \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F) for all hypergraphs having a special vertex contained in every edge. Given a graph , let us call the hypergraph obtained from by adding a new vertex having all triples with as edges the cone over , denoted by . So and the question is what one can say about \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(CG) in general. This problem is already very interesting when is a clique. Concerning stars the proof in [RRS-a] shows more generally that
[TABLE]
holds for all , but it remains unclear at this moment whether this is sharp for any . The (\frac{1}{3},\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense palette
[TABLE]
over establishes the lower bound \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(S_{4})\geq\frac{1}{3} and a generalisation of this idea leads to
[TABLE]
for all (see [RRS-a]*Section 5.3.1).
Example 2.4**.**
RĂśdlâs hypergraph, let us recall, is defined by the (\frac{1}{2},\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense palette
[TABLE]
over and establishes \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(K^{(3)}_{4})\geq\frac{1}{2}. More generally, given a set consisting of colours one may use the palette
[TABLE]
for showing
[TABLE]
It would be exciting if equality turned out to hold here for all . It should be pointed out, however, that if this is true it might be much more difficult to prove than Conjecture 1.3, as for there is a second, apparently sporadic, construction that yields the lower bound \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(K^{(3)}_{6})\geq\frac{3}{4} as well. Namely, one takes the palette over containing all six colour patterns involving both colours (see [RRS-a]*Section 5.1). This construction works because of . However, we are probably just exploiting a numerical coincidence here and it seems unlikely that similar Ramsey theoretic constructions are relevant to the problem of determining \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(K^{(3)}_{r+2}) (but see also Example 2.8).
Example 2.5**.**
Finally, we briefly discuss the case where is a cycle of length five, i.e., and E(C^{(3)}_{5})=\big{\{}\{i,i+1,i+2\}\colon i\in{\mathds{Z}}/5{\mathds{Z}}\bigr{\}}. The lower bound \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(C^{(3)}_{5})\geq\frac{4}{27} can be shown by using the set of colours and the palette consisting of all four colour patterns of the form , where ââ means either ââ or ââ. As far as we know, no interesting upper bound on \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(C^{(3)}_{5}) has ever been obtained.
The last example suggests that occasionally it may be more convenient to work with a weighted version of the concepts introduced so far. Let us say that a weighted set of colours is a pair consisting of a finite nonempty set of colours and a weight function with the property . If no weight function has been specified, we imagine that for all is implicitly understood. Now when we have a palette over such a weighted set of colours we say that is (d,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense if . In an obvious sense, this extends the meaning of being (d,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense introduced earlier. Now instead of artificially talking about dark and light red in Example 2.5 we could have just said that we consider the weighted set with and , as well as the palette .
As long as the values attained by our weight function are rational numbers, it remains, of course, purely a matter of taste whether one prefers weighted sets of colours or whether one rather wants to speak about different shades of colours that are somewhat immaterial to the definition of the palette. It is an interesting open question, however, whether allowing irrational weights of the colours can ever give rise to an optimal lower bound on \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F) for any hypergraph .
This roughly exhausts the lower bound constructions for \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(\cdot) that have been used so far, and we proceed with a discussion of \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(\cdot). Returning for simplicity to the unweighted setting, we say that a palette over a set of colours is (d,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense for a real number provided that
[TABLE]
Again easy probabilistic arguments show that whenever a palette is (d,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense, and a colouring gets chosen uniformly at random, then for every the hypergraph defined in (2.1) is asymptotically almost surely (d,\eta,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense. Thus lower bounds on \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F) can be established almost in the same way as for \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F), the only additional thing that needs to be checked being whether the palette one uses satisfies the three conditions in the above table.
For instance, the palettes we referred to in the Examples 2.3 and 2.4 are easily verified to be \mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}\,-dense for the expected values of . Hence the lower bounds on \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(\cdot) obtained there apply to \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(\cdot) as well. In particular, we learn
[TABLE]
for every and
[TABLE]
for every . But with the exception of Theorem 1.4 (and Theorem 1.2) it is not known whether equality holds here either. The reader might briefly wonder at this point whether \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F) and \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F) agree for all hypergraphs . But in unpublished work with RÜdl and Schacht it was shown that \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)>\pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)=0 holds for some hypergraph . Moreover, we obtained an explicit description of the class \{F\colon\pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)=0\}.
The story of \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(\cdot) starts similarly, but the few results that have been obtained so far seem to suggest that this generalised TurĂĄn density behaves quite differently. To begin with, given a real number and a palette over a set of colours , we say that is (d,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense if
[TABLE]
For clarity we emphasise that the two colours, , , etc. mentioned in the left column of this table are allowed to be identical. Now again standard probabilistic arguments show that if is (d,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense, then for every the hypergraph is asymptotically almost surely (d,\eta,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense and this principle can be used in the standard way for producing lower bounds on \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ 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}\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F) for many hypergraphs .
All palettes used in this connection so far are symmetrical in the sense that for every pattern and every permutation one has . In other words, this means that permuting the entries of a triple does not affect its membership in the palette. For symmetrical palettes any two of our three conditions are equivalent to each other, which reduces the amount of work one needs for checking them by a factor of three.111It is for this reason that in [RRS-d]*Section 13.1.3 only symmetrical palettes were introduced. Therefore, when writing [RRS-d], it seemed more convenient to define palettes as collections of multisets of colours instead of ordered triples, but it is unlikely that this will cause any confusion.
When specifying a symmetrical palette, it is convenient to enumerate only a small proportion of its colour patterns from which the remaining ones can be deduced owing to the symmetry condition. More precisely, given an arbitrary palette we call the inclusion-wise minimal symmetrical palette containing the symmetrical palette generated by . One may observe that the three symmetrical palettes in the examples that follow possess some further symmetries induced by permutations of colours.
Example 2.6**.**
The symmetrical palette over generated by
[TABLE]
is (\frac{1}{3},\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ 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}{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense and shows \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(K^{(3)}_{5})\geq\frac{1}{3} (see [RRS-d]*Section 13.1.3).
Example 2.7**.**
Similarly, the symmetrical palette over generated by is (\frac{1}{2},\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense and, due to the well-known Ramsey theoretic fact , this proves that \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(K^{(3)}_{6})\geq\frac{1}{2}.
Example 2.8**.**
Finally, the symmetrical palette over generated by
[TABLE]
is (\frac{2}{3},\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense and because of a Ramsey theoretic result due to Chung and Graham [CG83] this proves \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(K^{(3)}_{11})\geq\frac{2}{3}.
The main result of [RRS-d] provides an upper bound on the \mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}\,-Turån-densities of cliques that turns out to be sharp in surprisingly many small cases.
Theorem 2.9**.**
For every integer one has \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(K_{2^{r}})\leq\frac{r-2}{r-1}.
Together with the Examples 2.6â2.8 this yields
[TABLE]
i.e., the exact value of \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(K^{(3)}_{t}) for all with the exception of . It seems likely that if \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(K^{(3)}_{5})=\tfrac{1}{3} turned out to be true, then the methods of [RRS-d] would allow to prove \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(K^{(3)}_{10})\leq\tfrac{3}{5} as well. More generally, there are some good reasons to believe that \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(K^{(3)}_{\ell})=\alpha implies \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(K^{(3)}_{2\ell})\leq\frac{1}{2-\alpha}.
3. Reduced hypergraphs
It is currently open whether all extremal hypergraphs for \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}, \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}, and \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}} can be derived from palettes, i.e., whether they are of the form . There is, however, a slightly more general method to construct -dense hypergraphs with \star\in\{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ 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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}\}, for which such a result can be proved. This construction relies on so-called reduced hypergraphs that are going to be introduced next.
The main new idea is that when we have an ordered vertex set as well as a colouring of the pairs in , then in hypergraphs of the form the presence or absence of a triple with in depends entirely on the colours received by the pairs , , and without taking the relative positions of , , and in the linear ordering into account. But one could imagine, for instance, hypergraphs with vertex set for some huge , where for converting colour patterns observed on pairs into edges there is one rule applying to triples with two vertices in and a completely different rule for triples with two vertices in .
Reduced hypergraphs can be thought of as a framework for capturing the combinatorial core of all such constructions. Let us consider a finite set of indices. Suppose that to any pair of distinct indices there has been assigned a finite nonempty set of vertices , and that for distinct pairs of indices the corresponding vertex sets are disjoint. Finally, assume that for every triple of indices there has been specified a -uniform tripartite hypergraph with vertex classes , , and . In such situations we call the -partite -uniform hypergraph with
[TABLE]
a reduced hypergraph with index set , vertex classes , and constituents .
When translating such a reduced hypergraph into a hypergraph of TurĂĄn theoretic significance one starts with a huge vertex set having an equipartition V=\mathop{\vphantom{\bigcup}\mathchoice{\leavevmode\vtop{ \halign{\hfil\m@th\displaystyle#\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\textstyle#\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptstyle#\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptscriptstyle#\hfil\cr\bigcup\cr\cdot\crcr}}}}\displaylimits_{i\in I}V_{i} and takes a âcolouringâ of pairs of vertices such that for and with the pair receives uniformly at random some element of as its âcolourâ . Then for any three vertices from distinct partition classes , , and one decides whether should be the case depending on the colours , , and by using the constituent as if it were a palette; so explicitly one demands
[TABLE]
Next we need to express our density conditions in terms of reduced hypergraphs. The definition that follows is easy to remember. 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}\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}} dismissed at the the end of Section 1.3 would correspond to a minimum triple degree condition or, in other words, to the condition that all constituents be complete tripartite hypergraphs (if ). This is, of course, related to the fact that \mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}-dense hypergraphs of positive density contain everything, i.e., that \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)=0 holds for every hypergraph .
Definition 3.1**.**
Let denote a reduced hypergraph with index set , vertex classes , and constituents , and let be a real number.
- (*â*)
If holds for any three distinct indices we say that is (d,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense. 2. (*â*)
Moreover, if for any three distinct indices and every vertex we have
[TABLE]
then is called (d,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense. 3. (*â*)
Finally, if for any three distinct indices and all vertices , we have
[TABLE]
then is called (d,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense.
Whether the hypergraphs described by a given reduced hypergraph are capable of containing a given hypergraph can be expressed in terms of the existence of so-called reduced maps, that are going to be introduced next.
Definition 3.2**.**
A reduced map from a hypergraph to a reduced hypergraph with index set , vertex classes , and constituents is a pair such that
- (*â*)
and , where denotes the set of all pairs of vertices covered by an edge of ; 2. (*â*)
if , then and ; 3. (*â*)
if , then .
If some such reduced map exists, we say that contains a reduced image of , and otherwise is called -free.
Now the main result about the reduced hypergraph construction asserts the following.
Theorem 3.3**.**
If is a hypergraph and \star\in\{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}},\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}},\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ 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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}\}, then
[TABLE]
Large parts of the proof of this result are implicit in [RRS-a, RRS-c, RRS-d, RRS-zero]. Still, we believe it to be useful to gather the argument in its entirety in the remainder of this section and the two subsequent sections. To this end, we shall temporarily denote the right side of (3.3) by , where the superscript ââ means âreducedâ.
The inequality , proved in Proposition 3.4 below, simply expresses the fact that the narrative of this section does indeed indicate a valid strategy for establishing lower bounds on by means of reduced hypergraphs. The proof of the other direction, , requires more involved reasoning based on the hypergraph regularity method.
Proposition 3.4**.**
For every hypergraph and every symbol \star\in\{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ 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}\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope 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}{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}\} we have
[TABLE]
Let us recall the following standard concepts and facts required in the proof. A bipartite graph G=(X\mathbin{\mathchoice{\leavevmode\vtop{ \halign{\hfil\m@th\displaystyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\textstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptscriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}}Y,E) is called -quasirandom for two real numbers and if for all and the estimate \big{|}e(A,B)-d|A||B|\big{|}\leq\delta|X||Y| holds. Suppose now that for two nonempty disjoint sets and we create a random bipartite graph with vertex set X\mathbin{\mathchoice{\leavevmode\vtop{ \halign{\hfil\m@th\displaystyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\textstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptscriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}}Y by declaring each pair in uniformly at random to be an edge of with probability . Then for any fixed pair of sets and Chernoffâs inequality (see e.g. [AS]*Theorem A.1.4) implies
[TABLE]
whence
[TABLE]
In particular, if tends to infinity, then is asymptotically almost surely -quasirandom.
An important result about quasirandomness, utilised below, is the so-called triangle counting lemma. It informs us that if a tripartite graph P=(X\mathbin{\mathchoice{\leavevmode\vtop{ \halign{\hfil\m@th\displaystyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\textstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptscriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}}Y\mathbin{\mathchoice{\leavevmode\vtop{ \halign{\hfil\m@th\displaystyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\textstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptscriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}}Z,E) has the property that its naturally induced bipartite subgraphs on X\mathbin{\mathchoice{\leavevmode\vtop{ \halign{\hfil\m@th\displaystyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\textstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptscriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}}Y, X\mathbin{\mathchoice{\leavevmode\vtop{ \halign{\hfil\m@th\displaystyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\textstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptscriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}}Z, and Y\mathbin{\mathchoice{\leavevmode\vtop{ \halign{\hfil\m@th\displaystyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\textstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptscriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}}Z are -, -, and -quasirandom, respectively, then the size of the set of triangles it contains obeys the estimate
[TABLE]
Proof of Proposition 3.4.
Let a real number be given which has the property that for every there exists a -dense, -free reduced hypergraph with indices. We need to show that . So consider an arbitrary real as well as some . Now we need to produce a -dense, -free hypergraph with . For this purpose, we set
[TABLE]
and appeal to our hypothesis on . It yields an -free, -dense reduced hypergraph , say with index set , vertex classes , and constituents , where . Now set
[TABLE]
and let be sufficiently large.
We shall construct the desired hypergraph on a set of vertices V=\mathop{\vphantom{\bigcup}\mathchoice{\leavevmode\vtop{ \halign{\hfil\m@th\displaystyle#\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\textstyle#\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptstyle#\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptscriptstyle#\hfil\cr\bigcup\cr\cdot\crcr}}}}\displaylimits_{i\in I}V_{i} with for every . Owing to the probabilistic argument discussed before this proof we may assume that there is a family of colourings such that for every pair of indices and every the bipartite graph between and whose set of edges is happens to be -quasirandom. Depending on such colourings we complete the definition of in the expected way by setting
[TABLE]
Let us remark that all edges of are crossing in the sense of intersecting each of the vertex classes with at most once. The rationale behind our choice of in (3.58) is that it allows us to bound the number of non-crossing triples in a useful way. Clearly, this number is times the number of triples for which , , or holds. As this number is in turn at most , we conclude that the number of non-crossing ordered triples is at most , which by (3.58) is at most .
Now our choice of clearly guarantees . Next we would like to check that is indeed -free. Otherwise there would exist an embedding . For each let denote the index for which is true. For every pair we know that , because the edges of are crossing. Thus we may define by
[TABLE]
for every pair . Evidently and satisfy the first two clauses of Definition 3.2. As maps edges of to edges of , they satisfy (*â*) ⣠3.2 as well. Thus is a reduced map from to , contrary to the choice of as being -free.
It remains to check that is -dense and for this purpose we consider the three possibilities for separately.
First Case. \star=\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}
Given arbitrary we need to prove that |E_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(A,B,C)|\geq d|A||B||C|-\eta|V|^{3}. Whenever are distinct and , the triangle counting lemma entails that the tripartite subgraph of induced by , , and contains at least
[TABLE]
triangles each of which gives rise to an edge of . Thus for distinct we have
[TABLE]
which by our assumption that be (d,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense and by our choice of yields
[TABLE]
Summing over all ordered triples of distinct indices we infer that, up to an additive error of at most , the size of E_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(A,B,C) is at least times the number of crossing triples in . As there at most non-crossing triples altogether, it follows that we have indeed |E_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(A,B,C)|\geq d|A||B||C|-\eta|V|^{3}.
Second Case. \star=\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}
Given and we need to prove that |E_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(A,Q)|\geq d|A||Q|-\eta|V|^{3}. Getting rid of non-crossing triples as in the previous case, it suffices to this end if we show for any three distinct indices that
[TABLE]
For this in turn it is enough to establish that for every the sets
[TABLE]
and
[TABLE]
satisfy
[TABLE]
Now we distinguish the triples contributing to the left side according to the values of and . By the assumed (d,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-denseness of we know
[TABLE]
and thus it remains to show that for every edge of we have
[TABLE]
Now appealing for to the -quasirandomness of we learn that the sets
[TABLE]
and
[TABLE]
satisfy
[TABLE]
Summing over all we deduce
[TABLE]
Thus (3.67) will follow if can prove additionally that
[TABLE]
This estimate can be verified, however, in the same way as (3), the only difference being that this time one works with a sum over all and exploits the -quasirandomness of .
Third Case. \star=\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}
Proceeding almost exactly as in the previous case we consider two given sets of pairs and aiming at |E_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(Q,R)|\geq d|\mathcal{K}_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(Q,R)|-\eta|V|^{3} we begin again by eliminating the noncrossing triples from our consideration, this time by reducing our claim to the statement that for any three distinct indices the inequality
[TABLE]
holds. This will be clear once we know that for all and the sets
[TABLE]
and
[TABLE]
satisfy
[TABLE]
As is (d,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense, we have
[TABLE]
and, hence, it is enough to check that to every edge of the constituent there corresponds an inequality
[TABLE]
Now indeed for every the -quasirandomness of tells us that for the sets
[TABLE]
and
[TABLE]
one has
[TABLE]
By summing this over all one arrives at (3.69). â
4. Irregular triads
The definition of assures us that for every -dense reduced hypergraph with sufficiently many indices contains a reduced image of . For our intended application of this fact, however, we need to know that it remains true if one allows the deletion of a small number of edges from (see Proposition 4.4 below).
For \star=\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}} this turns out to be somewhat easier to prove than in the other two cases. The additional argument we want to put forth if \star=\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}} is the following.
Suppose that a reduced hypergraph with index set , vertex classes , and constituents as well as two real numbers and are given. We shall say that is (d,\eta,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense if for any three distinct indices the exceptional set consisting of all with
[TABLE]
satisfies . So for reduced hypergraphs being (d,0,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense means the same as being (d,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense.
Lemma 4.1**.**
For every hypergraph and every there exist and such that every (\pi^{\mathrm{rd}}_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)+\varepsilon,\eta,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense reduced hypergraph with indices contains a reduced image of .
Proof.
Choose in such a way that every \bigl{(}\pi^{\mathrm{rd}}_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)+\frac{\varepsilon}{2},\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope 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}\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}\bigr{)}-dense reduced hypergraph with indices contains a reduced image of , set
[TABLE]
and consider an arbitrary (\pi^{\mathrm{rd}}_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)+\varepsilon,\eta,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ 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}{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense reduced hypergraph with indices. As usual we denote the index set, vertex classes, and constituents of by , , and respectively. Let the exceptional sets be defined as above with \pi^{\mathrm{rd}}_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)+\varepsilon here in place of there.
Now the plan is to show that if one deletes all exceptional vertices from one gets a \bigl{(}\pi^{\mathrm{rd}}_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)+\frac{\varepsilon}{2},\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}\bigr{)}-dense reduced hypergraph, which, therefore, contains a reduced image of . Thus we define
[TABLE]
and notice that our assumption on implies
[TABLE]
whence, in particular, .
For this reason there exists a reduced hypergraph with index set and vertex classes whose constituents are the restrictions of to \mathcal{Q}^{ij}\mathbin{\mathchoice{\leavevmode\vtop{ \halign{\hfil\m@th\displaystyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\textstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptscriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}}\mathcal{Q}^{ik}\mathbin{\mathchoice{\leavevmode\vtop{ \halign{\hfil\m@th\displaystyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\textstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptscriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}}\mathcal{Q}^{jk}.
Consider any three distinct indices as well as an arbitrary vertex . From we conclude
[TABLE]
By (4.1) we have and , wherefore
[TABLE]
Combined with (4.18) this yields
[TABLE]
as desired. â
Similar considerations can be undertaken with respect to \mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}. As expected, a reduced hypergraph with standard notation is called (d,\eta,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense for two real numbers and provided that for any three distinct indices the set of all exceptional pairs with
[TABLE]
satisfies .
Lemma 4.2**.**
Given and a hypergraph , there are and such that every (\pi^{\mathrm{rd}}_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)+\varepsilon,\eta,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense reduced hypergraph with indices contains a reduced image of .
Proof.
Take so large that every \bigl{(}\pi^{\mathrm{rd}}_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)+\tfrac{\varepsilon}{2},\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}\bigr{)}-dense reduced hypergraph with indices contains a reduced image of . Take and fitting into the hierarchy
[TABLE]
and let be a (\pi^{\mathrm{rd}}_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)+\varepsilon,\eta,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ 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}\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense reduced hypergraph with index set of size , vertex classes , constituents , and exceptional sets (defined with \pi^{\mathrm{rd}}_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ 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}\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)+\varepsilon in place of ). We want to prove that there is a reduced map from to .
To this end we consider a random reduced hypergraph with index set whose vertex sets are any disjoint sets of size . The intended randomness is induced by a family of maps . Depending on the constituents of are defined so as to satisfy
[TABLE]
for all and all , , and .
Let us observe first that if for some choice of it happens that contains a reduced image of , then we are done. This is because if is a reduced map from to , then is a reduced map from to , where by we mean the map defined by (\psi\circ\varphi)(uv)=\psi^{\lambda(u)\lambda(v)}\bigl{(}\varphi(uv)\bigr{)} for all .
In the remainder of the proof we shall show that if gets chosen uniformly at random, then with positive probability is \bigl{(}\pi^{\mathrm{rd}}_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)+\tfrac{\varepsilon}{2},\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}\bigr{)}-dense, which will conclude the proof due to our choice of .
So let us study for fixed distinct indices and fixed vertices , the unpleasant event that the pair degree of and in is smaller than \bigl{(}\pi^{\mathrm{rd}}_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)+\tfrac{\varepsilon}{2}\bigr{)}\ell. If also the vertices and are given, then this pair degree depends only on and not on the remaining maps comprising . Moreover, the distribution of is the same as if one draws random elements from and keeps track of how many of them belong to the common neighbourhood of and in . Thus if the expected value of is at least (\pi^{\mathrm{rd}}_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)+\varepsilon)\ell and Chernoffâs inequality (see e.g. [AS]*Theorem A.1.4) yields
[TABLE]
Owing to we infer
[TABLE]
As there are altogether no more than possibilities to choose , , , , and , this proves
[TABLE]
and by an appropriate choice of and the right side can be pushed below . â
Remark 4.3**.**
It should be clear that the same construction could have been used for establishing Lemma 4.1. In fact, it generalises much further and applies to the study of the Turån densities initiated in [RRS-e] as well.
Proposition 4.4**.**
Given a hypergraph , a positive real number , and a symbol \star\in\{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}},\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}},\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}\} there exist and such that the following holds. If two reduced hypergraphs and with the same set of indices of size at least and with the same vertex classes have the properties that is -dense and
[TABLE]
then contains a reduced image of .
Proof.
We work with the hierarchy
[TABLE]
Call a triple useless if
[TABLE]
As a consequence of (4.19) the number of such useless triples is at most . Since is sufficiently large, we have and thus a proportion of no more than among all triples is useless. Therefore, if one draws a set with uniformly at random, the expected number of useless triples in is at most , which by an appropriate choice of can be made less than . For this reason, there exists a set with spanning no useless triple. We shall now prove that the restriction of to , denoted by in the sequel, contains a reduced image of . To this end we treat the three cases \star=\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ 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Since , the reduced hypergraph is \bigl{(}\pi^{\mathrm{rd}}_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)+\tfrac{\varepsilon}{2},\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}\bigr{)}-dense and thus it does indeed contain a reduced image of .
Second Case. \star=\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}
Owing to Lemma 4.1 it suffices to check that is \bigl{(}\pi^{\mathrm{rd}}_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)+\tfrac{\varepsilon}{2},2\xi\varepsilon^{-1},\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}\bigr{)}-dense. So let any three distinct indices be given and let denote the exceptional set of all vertices with
[TABLE]
Since is (\pi^{\mathrm{rd}}_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)+\varepsilon,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense and is useful, we have
[TABLE]
which yields indeed .
Third Case. \star=\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}
Arguing as the previous case one proves that is \bigl{(}\pi^{\mathrm{rd}}_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)+\tfrac{\varepsilon}{2},2\xi\varepsilon^{-1},\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}\bigr{)}-dense, which yields the desired conclusion in view of Lemma 4.2. â
5. Hypergraph regularity
The proof of Theorem 3.3 can now be completed by means of the hypergraph regularity method, which for -uniform hypergraphs is due to Frankl and RÜdl [FR]. Our presentation below also takes the later works [RoSchRL, RoSchCL, Gow06, NPRS09] into account.
A central notion in this area is that of a hypergraph being regular with respect to a tripartite graph , which roughly speaking means that the triangles in behave in an important way as if a random subset of them would correspond to edges of .
Definition 5.1**.**
A -uniform hypergraph is -regular with respect to a tripartite graph P=(X\mathbin{\mathchoice{\leavevmode\vtop{ \halign{\hfil\m@th\displaystyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\textstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptscriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}}Y\mathbin{\mathchoice{\leavevmode\vtop{ \halign{\hfil\m@th\displaystyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\textstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptscriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}}Z,E_{P}) with if for every tripartite subgraph we have
[TABLE]
Moreover, we simply say that * is -regular with respect to *, if it is -regular for some . We also define the relative density of with respect to by
[TABLE]
where we use the convention if .
Now the hypergraph regularity lemma tells us that large hypergraphs can in the following approximate sense be decomposed into regular parts.
Theorem 5.2** (Regularity Lemma).**
For every , every , and every there exists an integer such that for every and every -vertex -uniform hypergraph the following holds.
There are integers and , a vertex partition V_{0}\mathbin{\mathchoice{\leavevmode\vtop{ \halign{\hfil\m@th\displaystyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\textstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptscriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}}V_{1}\mathbin{\mathchoice{\leavevmode\vtop{ \halign{\hfil\m@th\displaystyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\textstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptscriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}}\dots\mathbin{\mathchoice{\leavevmode\vtop{ \halign{\hfil\m@th\displaystyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\textstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptscriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}}V_{t}=V, and for all there exists a partition
[TABLE]
of the edge set of the complete bipartite graph satisfying the following properties.
- (*â*)
* and ;* 2. (*â*)
for every and the bipartite graph is -regular; 3. (*â*)
and is -regular with respect to for all but at most tripartite graphs
[TABLE]
with and , , .
The tripartite graphs occurring in (5.97) are called triads. In order to get a better feeling as to why (in our context) such a decomposition of a given hypergraph is a useful thing to have, it may be helpful to imagine the following special outcome.
- (*â*)
, i.e., the entire vertex set gets partitioned; 2. (*â*)
every edge of intersects each partition class at most once; 3. (*â*)
there are no irregular triads, i.e., (*â*) ⣠5.2 holds without any exceptions; 4. (*â*)
moreover, all triads are either âfullâ in the sense that all their triangles correspond to edges of , or âemptyâ in the sense that none of their triangles correspond to edges of .
It is not hard to see that if these four things happen at the same time, then is essentially of the form constructed in the proof of Proposition 3.4. The underlying reduced hypergraph on which such a construction would be based has index set , vertex classes , and the possible edges in its constituents would indicate which triads are âfullâ.
So in a vague sense what remains to be done for completing the proof of Theorem 3.3 is that we need to address how to deal with the possible failures of (*â*) ⣠5â(*â*) ⣠5 when the regularity lemma gets applied. There will be no difficulties with (*â*) ⣠5 or (*â*) ⣠5, for the concepts we study are sufficiently robust, so that deleting the small set for (*â*) ⣠5 and ignoring the small proportion of noncrossing edges for (*â*) ⣠5 has essentially no effect. We are prepared for (*â*) ⣠5 in the light of Proposition 4.4.
Finally, regarding (*â*) ⣠5 we will treat triads with respect to which the relative density is not too small as if they were full. That is, for some appropriate constant we will put an edge into if and only if . This will allow us to rather easily transfer denseness properties from to , but we will need an argument as to why a reduced map from to does still give rise to a copy of in , even though the triads we want to use are not known to be full. This is, however, a standard situation in hypergraph regularity theory, for which the counting lemma has been developed. Below we shall require the following consequence of this result.
Theorem 5.3** (Embedding Lemma).**
For every -uniform hypergraph and every there exist , and functions and such that the following holds for every .
Suppose that
* is a map from to some set with for all ,* 2.
that is a family of mutually disjoint sets of the same size , 3.
and that for every one has a -quasirandom bipartite graph between and .
Then a hypergraph with posseses a subhypergraph isomorphic to provided that for every edge
one has 2.
and is -regular with respect to the tripartite graph .
For completeness we shall briefly discuss how this statement relates to the standard reference [NPRS09]*Corollary 2.3. First of all, a more conventional setup for the counting lemma would be the case that holds for some natural number and that is the identity. Secondly, in this special case the full counting lemma allows to estimate the number of homomorphisms from to with for every in a satisfactory way. In particular, a suitable choice of , , and entails that this number is at least . Thirdly, this assertion generalises immediately to the case of general , , and , even if should fail to be injective. Finally, by increasing if necessary, one can achieve that this lower bound on the number of homomorphisms from to exceeds the number of non-injective maps from to with for every . Therefore, [NPRS09]*Corollary 2.3 does indeed imply Theorem 5.3.
We may now proceed to the second half of Theorem 3.3.
Proposition 5.4**.**
If is a hypergraph and \star\in\{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}},\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}},\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}\}, then .
Proof.
We may suppose , since otherwise the result is clear. Let an arbitrary be given. By plugging and into Theorem 5.3 we obtain and functions as well as . Without loss of generality, we may suppose that is sufficiently small, that holds for every and that is increasing. By Proposition 4.4 and our flexibility to decrease we may assume that there exists such that if for arbitrary and one deletes deletes at most edges from a \bigl{(}\pi^{\mathrm{rd}}_{\star}(F)+\tfrac{1}{7}\varepsilon,\star\bigr{)}-dense reduced hypergraph with index set whose vertex classes have size , then the resulting reduced hypergraph contains a reduced image of . With this choice of , , and we appeal to the regularity lemma, thus getting an integer . Finally, we set
[TABLE]
Now we contend that every -dense hypergraph on vertices has a subhypergraph isomorphic to , which clearly implies the desired result.
Suppose that the regularity lemma applied to yields the integers and , the vertex partition V(H)=V_{0}\mathbin{\mathchoice{\leavevmode\vtop{ \halign{\hfil\m@th\displaystyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\textstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptscriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}}V_{1}\mathbin{\mathchoice{\leavevmode\vtop{ \halign{\hfil\m@th\displaystyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\textstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptscriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}}\dots\mathbin{\mathchoice{\leavevmode\vtop{ \halign{\hfil\m@th\displaystyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\textstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptscriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}}V_{t} and for the pair partition
[TABLE]
of such that (*â*) ⣠5.2, (*â*) ⣠5.2, and (*â*) ⣠5.2 hold.
This situation gives rise to two reduced hypergraphs and with index set and vertex classes for defined as follows. A triple is declared to form an edge of the constituent if the corresponding triad satisfies . If in addition is -regular with respect to this triad, then we put this edge into as well. We shall verify later that
[TABLE]
Based on this fact, the argument can be completed as follows. By Theorem 5.2(*â*) ⣠5.2 we have
[TABLE]
so due to our choice of according to Proposition 4.4 there is a reduced map from to . Now the embedding lemma applies to , , the sets for , and the bipartite graphs called here playing the rôles of there. The lower bound imposed there on the sets follows from
[TABLE]
for every . Moreover, satisfies the last two bullets of Theorem 5.3 by Definition 3.2(*â*) ⣠3.2 and the construction of . So altogether we obtain indeed and it remains to establish (5.130).
A key observation towards this goal is that for every triad spans at most \bigl{(}\ell^{-3}+3\delta_{2}(\ell)\bigr{)}M^{3} triangles due to the triangle counting lemma, and because of this is turn at most . So by our choice of a triad that does not correspond to an edge of can accomodate at most edges of .
Furthermore, it will be helpful to be aware that our choice of guarantees
[TABLE]
From now on we treat the three possibilities for separately.
First Case. \star=\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}
Given any three distinct indices we need to prove |E(\mathscr{A}^{ijk})|\geq\bigl{(}\pi^{\mathrm{rd}}_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)+\frac{1}{7}\varepsilon\bigr{)}\ell^{3}. Applying the assumption that is (\pi^{\mathrm{rd}}_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)+\varepsilon,\eta,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense to , , and we obtain
[TABLE]
Counting the edges of the left side according to the triad to which they belong we obtain
[TABLE]
Owing to
[TABLE]
this yields
[TABLE]
which is more than required.
Second Case. \star=\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}
Consider three distinct indices , a bipartite graph , and its neighbourhood
[TABLE]
in the constituent . Observe that
[TABLE]
Since is (\pi^{\mathrm{rd}}_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)+\varepsilon,\eta,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense, this yields
[TABLE]
where we have identified in the natural way with a subset of . As in the previous case this leads to
[TABLE]
which in turn implies
[TABLE]
Thus is indeed \bigl{(}\pi^{\mathrm{rd}}_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)+\tfrac{1}{7}\varepsilon,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ 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}{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ 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Third Case. \star=\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}
This time let three distinct indices as well as two bipartite graphs and be given, which we identify with the corresponding subsets of and , respectively. The graph counting lemma implies
[TABLE]
and it follows from being (\pi^{\mathrm{rd}}_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)+\varepsilon,\eta,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense that
[TABLE]
Regarding the common neighbourhood
[TABLE]
this tells us
[TABLE]
which yields
[TABLE]
as desired. â
6. More on tetrahedra
In order to illustrate how Theorem 3.3 can be applied we conclude this article by sketching a proof of \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(K^{(3)}_{4})=0. This result forms the first interesting case of Theorem 2.9 and the reader seeking further information or more details is referred to [RRS-d].
Given we want to show that every (\varepsilon,\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}})-dense reduced hypergraph with sufficiently many indices contains the reduced image of a tetrahedron. Let be such a reduced hypergraph with index set , vertex classes , and constituents . Write as a disjoint union I=X\mathbin{\mathchoice{\leavevmode\vtop{ \halign{\hfil\m@th\displaystyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\textstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{ \halign{\hfil\m@th\scriptscriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}}Y, where and is much larger then .
The first step is to assign to every pair an arbitrary vertex .
Next we look at two distinct vertices . For every the common neighbourhood of and in the constituent contains, by our hypothesis on , at least vertices. Thus, by double counting, we may fix a vertex belonging to this neighbourhood for at least many choices of . In other words, we may shrink by a factor of no more than to a subset such that is edge of for every .
This argument can be applied iteratively to all pairs of vertices in . That is, we enumerate all pairs in and when processing a pair in the list we select a vertex from the corresponding vertex class and shrink the subset of under current consideration by a further factor of . When this procedure ends, we have chosen for every pair a vertex . Moreover, if denotes the subset of that has survived through all stages, then holds for all distinct and all .
By starting with a sufficiently large set we can ensure that . Pick once and for all two distinct indices . Reversing the rôles of and we may now select a suitable vertex in and shrink in the same way as above to a set with such that holds for all . Due to there will be at least two survivors and in .
Now the four indices , , , and form together with the six vertices , , , , , and the desired reduced image of a tetrahedron in .
It should be clear that the same argument also establishes \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(B)=0 for every bipartite hypergraph . There are, however, many further hypergraphs whose \mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}-Turån-density vanishes. For instance, as a consequence of Theorem 2.2 the Fano plane satisfies {\pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(\mathscr{F})=0} and, hence, also \pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(\mathscr{F})=0. We shall return to the rather subtle problem of characterising the set \{F\colon\pi_{\mathord{\scaleobj{1.2}{\scalerel*{\leavevmode\hbox{\set@color \leavevmode\hbox to71.99pt{\vbox to63pt{\pgfpicture\makeatletter\hbox{\hskip 34.7995pt\lower-24.38501pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\ignorespaces\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{9.95863pt}{28.45276pt}\pgfsys@curveto{9.95863pt}{33.95282pt}{5.50006pt}{38.41139pt}{0.0pt}{38.41139pt}\pgfsys@curveto{-5.50006pt}{38.41139pt}{-9.95863pt}{33.95282pt}{-9.95863pt}{28.45276pt}\pgfsys@curveto{-9.95863pt}{22.9527pt}{-5.50006pt}{18.49413pt}{0.0pt}{18.49413pt}\pgfsys@curveto{5.50006pt}{18.49413pt}{9.95863pt}{22.9527pt}{9.95863pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@moveto{-14.68224pt}{-14.22638pt}\pgfsys@curveto{-14.68224pt}{-8.72632pt}{-19.14081pt}{-4.26775pt}{-24.64087pt}{-4.26775pt}\pgfsys@curveto{-30.14093pt}{-4.26775pt}{-34.5995pt}{-8.72632pt}{-34.5995pt}{-14.22638pt}\pgfsys@curveto{-34.5995pt}{-19.72644pt}{-30.14093pt}{-24.18501pt}{-24.64087pt}{-24.18501pt}\pgfsys@curveto{-19.14081pt}{-24.18501pt}{-14.68224pt}{-19.72644pt}{-14.68224pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{-24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@moveto{34.5995pt}{-14.22638pt}\pgfsys@curveto{34.5995pt}{-8.72632pt}{30.14093pt}{-4.26775pt}{24.64087pt}{-4.26775pt}\pgfsys@curveto{19.14081pt}{-4.26775pt}{14.68224pt}{-8.72632pt}{14.68224pt}{-14.22638pt}\pgfsys@curveto{14.68224pt}{-19.72644pt}{19.14081pt}{-24.18501pt}{24.64087pt}{-24.18501pt}\pgfsys@curveto{30.14093pt}{-24.18501pt}{34.5995pt}{-19.72644pt}{34.5995pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{24.64087pt}{-14.22638pt}\pgfsys@fillstroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{7.96677pt}\pgfsys@invoke{ }\ignorespaces{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{-24.64087pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{\ignorespaces}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@stroke@opacity{0}\pgfsys@invoke{ }\pgfsys@fill@opacity{0}\pgfsys@invoke{ }{}\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{1.5715pt}{35.71472pt}{2.84544pt}{34.14322pt}{2.84544pt}\pgfsys@curveto{32.57172pt}{2.84544pt}{31.29778pt}{1.5715pt}{31.29778pt}{0.0pt}\pgfsys@curveto{31.29778pt}{-1.5715pt}{32.57172pt}{-2.84544pt}{34.14322pt}{-2.84544pt}\pgfsys@curveto{35.71472pt}{-2.84544pt}{36.98866pt}{-1.5715pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\ignorespaces \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\ignorespaces}{\ignorespaces}{\ignorespaces}\hss}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{x}}}}(F)=0\} at another occasion.
Acknowledgement
For numerous reasons going much beyond our collaboration [RRS-a, RRS-c, RRS-d, RRS-e, RRS-zero] my indebtedness to VojtÄch RĂśdl and Mathias Schacht is extremely great.
References