The weak separation in higher dimensions

TL;DR

This paper introduces and explores the properties of weakly r-separated collections of subsets in higher dimensions, extending prior concepts and providing a geometric framework for their combinatorial analysis.

Contribution

It generalizes weak separation to higher dimensions and develops a geometric approach to analyze maximal collections, extending previous work by Leclerc and Zelevinsky.

Findings

Established combinatorial properties of maximal weakly r-separated collections

Extended weak separation concepts to higher dimensions and even r cases

Developed a geometric framework for analyzing these collections

Abstract

For an odd integer $r>0$ and an integer $n>r$, we introduce a notion of weakly $r$-separated collections of subsets of $[n]=\{1,2,\ldots,n\}$. When $r=1$, this corresponds to the concept of weak separation introduced by Leclerc and Zelevinsky. In this paper, extending results due to Leclerc-Zelevinsky, we develop a geometric approach to establish a number of nice combinatorial properties of maximal weakly r-separated collections. As a supplement, we also discuss an analogous concept when $r$ is even.

Equations

$$\mathcal{N}=\mathcal{N}(P,Q):=\{S\subset P\cup Q:S\ne P,Q,r^{\prime }\le |S|\le r^{\prime }+1\}.$$

$${\mathcal{W}}^{-}$$

$$\mathcal{T}$$

$$|\mathcal{T}|\le \left(\genfrac{}{}{0ex}{}{n-1}{\le r}\right).$$

$${\mathcal{D}}^i:={\mathcal{T}}^i\cup ({\mathcal{S}}^{i+1}\cup \dots \cup {\mathcal{S}}^{n-1})\cup ({\mathcal{A}}^0\cup {\mathcal{A}}^1\cup \dots \cup {\mathcal{A}}^{i-1}).$$

$$|{\mathcal{D}}^i|\le \left(\genfrac{}{}{0ex}{}{n-1}{\le r-1}\right)=s_{n-1,r-2}=|V(Z^{\mathrm{fr}})|.$$

$$|{\mathcal{T}}^i|=|{\mathcal{D}}^i|-(|V(Z^{\mathrm{fr}})-{\mathcal{S}}^{\prime }|)-|{\mathcal{A}}^{\prime }|\le |{\mathcal{S}}^{\prime }|-|{\mathcal{A}}^{\prime }|.$$

$$|{\mathcal{T}}^i|\le |{\mathcal{R}}^i|.$$

$$|\mathcal{T}|={\sum }_i|{\mathcal{T}}^i|\le {\sum }_i|{\mathcal{R}}^i|=|V_Q|=s_{n-1,r-1}=\left(\genfrac{}{}{0ex}{}{n-1}{\le r}\right),$$

$${\mathcal{N}}^{\uparrow }(P,Q)$$

$${\mathcal{N}}^{\downarrow }(P,Q)$$

$$(X|\left(\genfrac{}{}{0ex}{}{T}{h}\right))(=\{X\cup A:A\subset T,|A|=h\}).$$

$$x=(x_1,\dots ,x_d)\mapsto (x_1-ϵx_d,x_2,\dots ,x_{d-1})=:{\pi }^{ϵ}(x)$$

$${\psi }_i:=\pi ({\xi }_i)\text{mbox}and{\psi }_i^{ϵ}:={\pi }^{ϵ}({\xi }_i),i=1,\dots ,n.$$

$$V(\Delta )=(X|\left(\genfrac{}{}{0ex}{}{T}{h-1}\right))\cup (X|(\genfrac{}{}{0ex}{}{T}{h}))\text{mbox}andR\subseteq V({\Delta }^{\mathrm{fr}})\cup V({\Delta }^{\mathrm{rear}})$$

$$\mathcal{A}:=V(Z)\cup \{A_1,A_5,A_9\}.$$

$$\ell ({\Delta }_0)\le \ell ({\Delta }_1)\le \cdots \le \ell ({\Delta }_{k-1})\le \ell ({\Delta }_0).$$

Full text

The weak separation in higher dimensions

Vladimir I. Danilov Central Institute of Economics and Mathematics of the RAS, 47, Nakhimovskii Prospect, 117418 Moscow, Russia; email: danilov@cemi.rssi.ru.   

Alexander V. Karzanov Central Institute of Economics and Mathematics of the RAS, 47, Nakhimovskii Prospect, 117418 Moscow, Russia; email: akarzanov7@gmail.com. Corresponding author.   

Gleb A. Koshevoy The Institute for Information Transmission Problems of the RAS, 19, Bol’shoi Karetnyi per., 127051 Moscow, Russia; email: koshevoyga@gmail.com. Supported in part by grant RSF 16-11-10075.

Abstract. For an odd integer $r>0$ and an integer $n>r$, we introduce a notion of weakly $r$-separated collections of subsets of $[n]=\{1,2,\ldots,n\}$. When $r=1$, this corresponds to the concept of weak separation introduced by Leclerc and Zelevinsky. In this paper, extending results due to Leclerc-Zelevinsky, we develop a geometric approach to establish a number of nice combinatorial properties of maximal weakly r-separated collections.

As a supplement, we also discuss an analogous concept when $r$ is even.

Posted on arXiv:1904.09798[math.CO].

Keywords : weakly separated sets, cyclic zonotope, fine zonotopal tiling, higher Bruhat order

MSC Subject Classification  05E10, 05B45

1 Introduction

Let $n$ be a positive integer and let $[n]$ denote the set $\{1,2,\ldots,n\}$. For subsets $X,Y\subseteq[n]$, we write $X<Y$ if the maximal element $\max(X)$ of $X$ is smaller than the minimal element $\min(Y)$ of $Y$, letting $\max(\emptyset):=0$ and $\min(\emptyset):=n+1$. An interval in $[n]$ is a nonempty subset $\{a,a+1,\ldots,b\}$ in it, denoted as $[a,b]$ (so $[n]=[1,n]$).

The well-known concept of strongly separated sets introduced by Leclerc and Zelevinsky [8] is extended as follows.

Definition.  For $r\in{\mathbb{Z}}_{\geq 0}$, sets $A,B\subseteq[n]$ are called (strongly) $r$-separated if there is no sequence $i_{1}<i_{2}<\cdots<i_{r+2}$ of elements of $[n]$ such that the elements with odd indices (namely, $i_{1},i_{3},\ldots$) belong to one of $A-B$ and $B-A$, while the elements with even indices ($i_{2},i_{4},\ldots$) belong to the other of these two sets (where $A^{\prime}-B^{\prime}$ denotes the set difference $\{i\colon A^{\prime}\ni i\notin B^{\prime}\}$). Accordingly, a set-system ${\cal S}\subseteq 2^{[n]}$ (a collection of subsets of $[n]$) is called $r$-separated if any two members of ${\cal S}$ are such.

Equivalently, $A,B\subseteq[n]$ are $r$-separated if there are intervals $I_{1}<I_{2}<\cdots<I_{r^{\prime}}$ in $[n]$ with $0\leq r^{\prime}\leq r+1$ such that one of $A-B$ and $B-A$ is included in $I_{1}\cup I_{3}\cup\ldots$ , and the other in $I_{2}\cup I_{4}\cup\ldots$ . If, in addition, $r^{\prime}+|I_{1}|+\cdots+|I_{r^{\prime}}|$ is as small as possible, we say that $(I_{1},\ldots,I_{r^{\prime}})$ is the interval cortege associated with $A,B$.

In particular, $A,B$ are 0-separated if $A\subseteq B$ or $B\subseteq A$, and 1-separated if either $\max(A-B)<\min(B-A)$ or $\max(B-A)<\min(A-B)$. The 1-separation relation is just what is called the strong separation one in [8]. The case $r=2$ was studied by Galashin [6] (who used the term “chord separated” for 2-separated sets). A study for a general $r$ is conducted in Galashin and Postnikov [7].

When $A,B$ are $r$-separated but not $(r-1)$-separated, they are called $(r+1)$-interlaced. In other words, the interval cortege associated with such $A,B$ consists of $r+1$ intervals. For example, $A=\{1,2,5,6,7,10\}$ and $B=\{2,3,6,9\}$ have the interval cortege $(\{1\},\,\{3\},\,[5,7],\,\{9\},\{10\})$), and therefore they are 5-interlaced.

Another sort of set separation introduced by Leclerc and Zelevinsky is known under the name of weak separation (which appeared in [8] in connection with the problem of characterizing quasi-commuting flag minors of a quantum matrix; for a discussion on this and wider relations between the weak separation and quantum minors, see also [1, Sect. 8]). We generalize that notion to “higher dimensions” in the following way (where the term “higher dimensions” is justified by appealing to a geometric interpretation, defined later). When $A,B\subseteq[n]$ are such that $\min(A-B)<\min(B-A)$ and $\max(A-B)>\max(B-A)$, we say that $A$ surrounds $B$.

Definition.  Let $r$ be a positive odd integer. Sets $A,B\subseteq[n]$ are called weakly $r$-separated if they are $r^{\prime}$-interlaced with $r^{\prime}\leq r+2$, and if $r^{\prime}=r+2$ takes place, then either (a)  $A$ surrounds $B$ and $|A|\leq|B|$, or (b)  $B$ surrounds $A$ and $|B|\leq|A|$. Accordingly, a set-system ${\cal W}\subseteq 2^{[n]}$ is called weakly $r$-separated if any two members of ${\cal W}$ are such.

In other words, $A$ and $B$ are weakly $r$-separated if they are either (strongly) $r$-separated or $(r+2)$-interlaced, and in the latter case, for the interval cortege $(I_{1},\ldots,I_{r+2})$ associated with $A,B$, if the cardinatilies of $A$ and $B$ are different, say, $|A|<|B|$, then $I_{1}\cup I_{3}\cup\ldots\cup I_{r+2}$ contains $A-B$ (and $I_{2}\cup I_{4}\cup\ldots\cup I_{r+1}$ contains $B-A$). For example, $\{1,2,6\}$ and $\{2,3,4,5\}$ are weakly 1-separated, whereas $\{1,2,5,6,7\}$ and $\{1,3,4,5\}$ are 3-interlaced (having the interval cortege $(\{2\},[3,4],[6,7])$) but not weakly 1-separated.

When $r=1$, the notion of weak 1-separation turns into the weak separation of [8].

In this paper we generalize, to an arbitrary odd $r\geq 1$, two results on weakly separated collections obtained in [8]. One of those says that

1. (1.1)

the maximal possible sizes (numbers of members) of strongly and weakly separated collections in $2^{[n]}$ are the same and equal to $\frac{1}{2}n(n+1)+1$ ($=\binom{n}{2}+\binom{n}{1}+\binom{n}{0}$).

To formulate a generalization of ((1.1)), let $r<n$ and denote the maximal possible size $|{\cal S}|$ of an $r$-separated collection ${\cal S}\subseteq 2^{[n]}$ by $s_{n,r}$. Also when $r$ is odd, denote the maximal possible size of a weakly $r$-separated collection ${\cal W}\subseteq 2^{[n]}$ by $w_{n,r}$. Extending results in [8] (for $r=1$) and [6] (for $r=2$), Galashin and Postnikov [7] showed that

1. (1.2)

$s_{n,r}=\binom{n}{\leq r+1}$   $\left(=\binom{n}{r+1}+\binom{n}{r}+\cdots+\binom{n}{0}\right)$.

We prove the following

Theorem 1.1

Let $r$ be odd. Then $w_{n,r}=s_{n,r}$.

Another impressive result in [8] says that a weakly separated collection can be transformed into another one by making a flip (a sort of mutations) “in the presence of four witnesses”. This relies on the following property (Theorem 7.1 in [8]):

1. (1.3)

let ${\cal W}\subset 2^{[n]}$ be weakly separated, and suppose that there are elements $i<j<k$ of $[n]$ and a set $X\subseteq[n]-\{i,j,k\}$ such that ${\cal W}$ contains four sets (“witnesses”) $Xi$, $Xk$, $Xij$, $Xjk$ and a set $U\in\{Xj,Xik\}$; then the collection obtained from ${\cal W}$ by replacing $U$ by the other member of $\{Xj,Xik\}$ is again weakly separated.

Hereinafter, for disjoint subsets $A$ and $\{a,\ldots,b\}$ of $[n]$, we use the abbreviated notation $Aa\ldots b$ for $A\cup\{a,\ldots,b\}$.

We generalize ((1.3)) as follows.

Theorem 1.2

For an odd $r$ and $r^{\prime}:=(r+1)/2$, let $P=\{p_{1},\ldots,p_{r^{\prime}}\}$ and $Q=\{q_{0},\ldots,q_{r^{\prime}}\}$ consist of elements of $[n]$ such that $q_{0}<p_{1}<q_{1}<p_{2}<\ldots<p_{r^{\prime}}<q_{r^{\prime}}$, and let $X\subseteq[n]-(P\cup Q)$. Define the set of neighbors (or “witnesses”) of $P,Q$ to be

[TABLE]

Suppose that a weakly $r$-separated collection ${\cal W}\subset 2^{[n]}$ contains a set $U\in\{X\cup P,\,X\cup Q\}$. If, in addition, ${\cal W}$ contains the sets of the form $X\cup S$ for all $S\in{\cal N}$, then the collection obtained from ${\cal W}$ by replacing $U$ by the other member $U^{\prime}$ of $\{X\cup P,\,X\cup Q\}$ is weakly $r$-separated as well.

(Obviously, $P$ and $Q$ are not weakly $r$-separated, and $|P\cup Q|=r+2$ easily implies that any two sets in ${\cal N}\cup\{P,Q\}$ except for $P,Q$ are weakly $r$-separated.) In general, for two weakly $r$-separated collections ${\cal W}$ and ${\cal W}^{\prime}$, if there are $P,Q,X$ as above such that ${\cal W}^{\prime}=({\cal W}-\{X\cup P\})\cup\{X\cup Q\}$ and ${\cal W}=({\cal W}^{\prime}-\{X\cup Q\})\cup\{X\cup P\}$, then we say that ${\cal W}^{\prime}$ is obtained from ${\cal W}$ by a raising (combinatorial) flip, while ${\cal W}$ is obtained from ${\cal W}^{\prime}$ by a lowering flip.

Our method of proof of the above theorems (and more) appeals to a geometric approach and uses some facts on fine zonotopal tilings, or cubillages, on a cyclic zonotope in a space ${\mathbb{R}}^{d}$. One of them is that the maximal by size (strongly) $(d-1)$-separated collections ${\cal S}$ in $2^{[n]}$ one-to-one correspond to the cubillages $Q$ in a cyclic zonotope $Z(n,d)$ generated by (a cyclic configuration of) $n$ vectors in ${\mathbb{R}}^{d}$; one may say that the set of vertices of $Q$ “encodes” ${\cal S}$. (When $d=2$, a cubillage becomes a rhombus tiling on a planar $n$-zonogon, and a bijection between these tilings and the maximal strongly separated collections in $2^{[n]}$ is well-known. For $d=3$, a bijection between the corresponding cubillages and maximal 2-separated sets was originally established in [6]. For a general $d$, the corresponding bijection was recently shown by Galashin and Postnikov [7].)

Another useful fact, inspired by a result in the classical work due to Manin and Schechtman [9] on higher Bruhat orders, is that any cubillage on $Z(n,d-1)$ can be lifted as a certain $(d-1)$-dimensional subcomplex, that we call an s-membrane, in some cubillage on $Z(n,d)$. For more details and other relevant facts, see [5, 10].

We further develop the theory of cubillages by constructing a certain fragmentation $Q^{\equiv}$ of a cubillage $Q$ on $Z(n,d)$, introducing a class of $(d-1)$-dimensional subcomplexes in $Q^{\equiv}$, called w-membranes, and showing (in Theorem 6.4) that when $d$ is odd, the vertex set of any w-membrane forms a maximal by size weakly $(d-2)$-separated collection in $2^{[n]}$. It turns out that the collections of this sort (over all cubillages on $Z(n,d)$) constitute a poset with a unique minimal element and a unique maximal element and where neighboring collections are linked by flips; this is obtained as a consequence of Theorems 6.4 and 1.2.

In light of this, given an odd $r$ and $n>r$, we can specify three classes ${\bf W}_{n,r}$, ${\bf W}^{=}_{n,r}$ and ${\bf W}^{\ast}_{n,r}$ of weakly $r$-separated collections ${\cal W}$ in $2^{[n]}$, in which ${\cal W}$ is maximal by inclusion, maximal by size, and representable, respectively. Here we call ${\cal W}$ representable if it can be represented as the vertex set of a w-membrane in a cubillage on $Z(n,r+2)$ (in particular, ${\cal W}$ is maximal by size). Then ${\bf W}_{n,r}\supseteq{\bf W}^{=}_{n,r}\supseteq{\bf W}^{\ast}_{n,r}$.

This paper is organized as follows. Sect. 2 contains basic definitions and reviews some useful facts on cyclic zonotopes and cubillages. Sect. 3 recalls the construction of s-membranes in cubillages and describes their properties needed to us. Here we also introduce the so-called bead-thread relation on vertices of a cubillage, which is used in the proof of Theorem 1.1. Sect. 4 is devoted to proving Theorem 1.1, and Sect. 5 proves a sharper version of Theorem 1.2 (given in Theorem 5.1 and Corollary 5.3 where instead of the whole set ${\cal N}(P,Q)$ of neighbors of $P,Q$ we take into account only those neighbors that are at distance $\leq 2$ from $P$ or $Q$).

Sect. 6 introduces w-membranes in the fragmentation of a cubillage and proves the above-mentioned results on w-membranes in a cubillage on $Z(n,d)$ and representable $(d-2)$-separated collection in $2^{[n]}$, and on the poset of such collections (Theorem 6.4 and Corollary 6.5). Sect. 7 demonstrates an example of “non-pure” weakly $r$-separated collections ${\cal W}$ in $2^{[n]}$, in the sense that ${\cal W}\in{\bf W}_{n,r}$ but $|{\cal W}|<w_{n,r}$, thus showing that the inclusion ${\bf W}_{n,r}\supseteq{\bf W}^{=}_{n,r}$ can be strict (in contrast to the well-known purity result for $r=1$, saying that ${\bf W}_{n,1}={\bf W}^{=}_{n,1}$). Also we raise two conjectures on weakly $r$-separated set-systems (in Sects. 6 and 7).

Proofs of two propositions stated in Sect. 6 are given in Appendix A. The paper finishes with Appendix B where we discuss a reasonable analog of the weak $r$-separation when $r$ is even, outline some constructions and results on this way and raise two more conjectures.

2 Preliminaries

This section contains additional definitions, notation and conventions that will be needed later on. Also we review some known properties of cubillages.

$\bullet$  Let $n,d$ be positive integers with $n\geq d>1$. By a cyclic configuration of size $n$ in ${\mathbb{R}}^{d}$ we mean an ordered set $\Xi$ of $n$ vectors $\xi_{i}=(\xi_{i}(1),\ldots,\xi_{i}(d))\in{\mathbb{R}}^{d}$, $i=1,\ldots,n$, satisfying:

1. (2.1)

• (a)

$\xi_{i}(1)=1$ for each $i$, and

• (b)

for the $d\times n$ matrix $A$ formed by $\xi_{1},\ldots,\xi_{n}$ as columns (in this order), any flag minor of $A$ is positive.

A typical (and commonly used) sample of such configurations $\Xi$ is generated by the Veronese curve; namely, take reals $t_{1}<t_{2}<\cdots<t_{n}$ and assign $\xi_{i}:=\xi(t_{i})$, where $\xi(t)=(1,t,t^{2},\ldots,t^{d-1})$.

The zonotope $Z=Z(\Xi)$ generated by $\Xi$ is the Minkowski sum of line segments $[0,\xi_{i}]$, $i=1,\ldots,n$. A fine zonotopal tiling is a subdivision $Q$ of $Z$ into $d$-dimensional parallelotopes such that: any two intersecting ones share a common face, and each face of the boundary of $Z$ is contained in some of these parallelotopes. For brevity, we liberally refer to these parallelotopes as cubes, and to $Q$ as a cubillage.

$\bullet$  When $n,d$ are fixed, the choice of one or another cyclic configuration $\Xi$ (subject to ((2.1))) does not matter in essence, and for this reason, we unify notation $Z(n,d)$ for $Z(\Xi)$, referring to it as the cyclic zonotope for $(n,d)$.

$\bullet$  Let $\pi$ denote the projection ${\mathbb{R}}^{d}\to{\mathbb{R}}^{d-1}$ given by $(x_{1},\ldots,x_{d})\mapsto(x_{1},\ldots,x_{d-1})$. Due to ((2.1)), the vectors $\pi(\xi_{1}),\ldots,\pi(\xi_{n})$ form a cyclic configuration as well, and we may say that $\pi$ projects $Z(n,d)$ to the zonotope $Z(n,d-1)$.

$\bullet$  Each subset $X\subseteq[n]$ naturally corresponds to the point $\sum_{i\in X}\xi_{i}$ in $Z(n,d)$, and the cardinality $|X|$ is called the height, or level of this subset/point. (W.l.o.g., we usually assume that all combinations of vectors $\xi_{i}$ with coefficients 0,1 are different.)

$\bullet$  Depending on the context, we may think of a cubillage $Q$ on $Z(n,d)$ in two ways: either as a set of $d$-dimensional cubes (and write $C\in Q$ for a cube $C$ in $Q$) or as the corresponding polyhedral complex. The 0- and 1-dimensional cells (or faces) of $Q$ are called vertices and edges, respectively. A simple fact is that, by the subset-to-point correspondence, each vertex is identified with a subset of $[n]$. In turn, each edge $e$ is a parallel translation of some segment $[0,\xi_{i}]$; we say that $e$ has color $i$, or is an $i$-edge. When needed, $e$ is regarded as a directed edge (according to the direction of $\xi_{i}$).

$\bullet$  Let $V(Q)$ denote the set of vertices of a cubillage $Q$. Galashin and Postnikov [7] showed the following important correspondence between cubillages and separated set-systems:

1. (2.2)

for any cubillage $Q$ on $Z(n,d)$, the set $V(Q)$ of its vertices (regarded as subsets of $[n]$) constitutes a maximal by size $(d-1)$-separated collection in $2^{[n]}$; conversely, for any maximal by size $(d-1)$-separated collection ${\cal S}\subseteq 2^{[n]}$, there exists a cubillage $Q$ on $Z(n,d)$ with $V(Q)={\cal S}$.

$\bullet$  When a cell (face) $C$ of $Q$ has the lowest point $X\subseteq[n]$ and when $T\subseteq[n]$ is the set of colors of edges in $C$, we say that $C$ has the root $X$ and type $T$, and may write $C=(X\,|\,T)$. One easily shows that $X\cap T=\emptyset$. Another well-known fact is that for any cubillage $Q$, the types of all ($d$-dimensional) cubes in it are different and form the set $\binom{[n]}{d}$ of $d$-element subsets of $[n]$ (so $Q$ has exactly $\binom{n}{d}$ cubes).

$\bullet$  For a closed subset $U$ of points in $Z=Z(n,d)$, let $U^{\rm fr}$ ($U^{\rm rear}$) be the part of $U$ “seen” in the direction of the last, $d$-th, coordinate vector $e_{d}$ (resp. $-e_{d}$), i.e., the set formed by the points $x\in\pi^{-1}(x^{\prime})\cap U$ with $x_{d}$ minimum (resp. maximum) for all $x^{\prime}\in\pi(U)$. It is called the front (resp. rear) side of $U$.

In particular, $Z^{\,\rm fr}$ and $Z^{\,\rm rear}$ denote the front and rear sides, respectively, of (the boundary of) the zonotope $Z$. We call $Z^{\,\rm fr}\cap Z^{\,\rm rear}$ the rim of $Z$ and denote it as $Z^{\,\rm rim}$.

$\bullet$  When a set $X\subseteq[n]$ is the union of $k$ intervals and $k$ is as small as possible, we say that $X$ is a $k$-interval. Note that for such an $X$, its complementary set $[n]-X$ is a $k^{\prime}$-interval with $k^{\prime}\in\{k-1,k,k+1\}$. In the next section we will use the following known characterization of the sets of vertices in the front and rear sides of a zonotope of an odd dimension (cf., e.g., [5]).

1. (2.3)

Let $d$ be odd. Then for $Z=Z(n,d)$,

• (i)

$V(Z^{\,\rm fr})$ is formed by all $k$-intervals of $[n]$ with $k\leq(d-1)/2$;

• (ii)

$V(Z^{\,\rm rear})$ is formed by the subsets of $[n]$ complementary to those in $V(Z^{\,\rm fr})$; specifically, it consists of all $k$-intervals with $k<(d-1)/2$, all $(d-1)/2$-intervals containing at least one of the elements 1 and $n$ and all $(d+1)/2$-intervals containing both 1 and $n$.

This implies that $V(Z^{\,\rm rim})$ consists of the $k$-intervals with $k<(d-1)/2$ and the $(d-1)/2$-intervals containing at least one of 1 and $n$; the set of inner vertices in $Z^{\,\rm fr}$, i.e., $V(Z^{\,\rm fr})-V(Z^{\,\rm rim})$ consists of the $(d-1)/2$-intervals containing none of 1 and $n$, whereas $V(Z^{\,\rm rear})-V(Z^{\,\rm rim})$ consists of the $(d+1)/2$-intervals containing both 1 and $n$.

$\bullet$   Consider a cube $C=(X\,|\,T)$ and let $T=(p(1)<p(2)<\cdots<p(d))$. This cube has $2d$ facets $F_{1},\ldots,F_{d},G_{1},\ldots,G_{d}$, where

1. (2.4)

$F_{i}=F_{i}(C)$ is viewed as $(X\,|\,T-p(i))$, and $G_{j}=G_{j}(C)$ as $(Xp(j)\,|\,T-p(j))$.

(For a set $A$ and an element $a\in A$, we abbreviate $A-\{a\}$ to $A-a$.)

3 S-membranes and bead-threads

In this section we recall the definition of s-membranes, associate with a cubillage a certain path structure, and review some basic properties.

Definition.  Let $Q$ be a cubillage on $Z(n,d)$. An s-membrane in $Q$ is a subcomplex $M$ of $Q$ such that $M$ (regarded as a subset of ${\mathbb{R}}^{d}$) is bijectively projected by $\pi$ to $Z(n,d-1)$.

Then each facet of $Q$ occurring in $M$ is projected to a cube of dimension $d-1$ in $Z(n,d-1)$ and these cubes constitute a cubillage on $Z(n,d-1)$, denoted as $\pi(M)$. In view of ((2.2)) and ((1.2)) (applied to $\pi(Q)$), we obtain that

1. (3.1)

all s-membranes $M$ in a cubillage $Q$ on $Z(n,d)$ have the same number of vertices, which is equal to $s_{n,d-2}$, and the vertex set of $M$ (regarded as a collection in $2^{[n]}$) is $(d-2)$-separated.

Two s-membranes are of an especial interest. These are the front side $Z^{\,\rm fr}$ and the rear side $Z^{\,\rm rear}$ of $Z=Z(n,d)$ (in these cases the choice of a cubillage on $Z$ is not important.) Following terminology in [4, 5], their projections $\pi(Z^{\,\rm fr})$ and $\pi(Z^{\,\rm rear})$ are called the standard and anti-standard cubillages on $Z(n,d-1)$, respectively.

Next we distinguish certain vertices in cubes. When $n=d$, the zonotope turns into the cube $C=(\emptyset|[d])$, and there holds:

1. (3.2)

the front side $C^{\,\rm fr}$ (rear side $C^{\,\rm rear}$) of $C=(\emptyset|[d])$ has a unique inner vertex, namely, $t_{C}:=\{i\in[n]\colon d-i$ odd$\}$ (resp. $h_{C}:=\{i\in[n]\colon d-i$ even$\}$.

(When $d$ is odd, ((3.2)) can be obtained from ((2.3)). A direct proof of ((3.2)) for an arbitrary $d$ is as follows (a sketch). The facets of $C$ are $F_{i}=F_{i}(C):=(\emptyset|[d]-i)$ and $G_{i}=G_{i}(C):=(i|[d]-i)$, $i=1,\ldots,d$ (cf. ((2.4))). A facet $F_{i}$ is contained in $C^{\,\rm fr}$ ($C^{\,\rm rear}$) if, when looking at the direction $e_{d}$, $C$ lies “behind” (resp. “before”) the hyperplane containing $F_{i}$, or, equivalently, $\det(A_{i})>0$ (resp. $\det(A_{i})<0$), cf. ((2.1))(b), where $A_{i}$ is the matrix with the columns $\xi_{1},\ldots,\xi_{i-1},\xi_{i+1},\ldots\xi_{d},\xi_{i}$ (in this order). It follows that $F_{i}\subset C^{\,\rm fr}$ if and only if $d-i$ is even. By “central symmetry”, $G_{i}\subset C^{\,\rm fr}$ if and only if $d-i$ is odd.

Now consider a vertex $X\subseteq[d]$ of $C$. If $X$ (resp. $[d]-X$) has consecutive elements $i-1$ and $i$, then $X\in G_{i-1}$ and simultaneously $X\in G_{i}$ (resp. $X\in F_{i-1}$ and $X\in F_{i}$). This implies that $X$ is in both $C^{\,\rm fr}$ and $C^{\,\rm rear}$, i.e., $X\in C^{\,\rm rim}$. The remaining vertices of $C$ are just $t_{C}$ and $h_{C}$ as in ((3.2)); one can see that the former (latter) is contained in all facets $F_{j}$ and $G_{i}$ with $d-j$ even and $d-i$ odd (resp. $d-j$ odd and $d-i$ even). So $t_{C}$ lies in $C^{\,\rm fr}$, and $h_{C}$ in $C^{\,\rm rear}$; moreover, both are not in $C^{\,\rm rim}$ (since $C$ is full-dimensional).)

When $n$ is arbitrary and $Q$ is a cubillage on $Z=Z(n,d)$, we distinguish vertices $t_{C}$ and $h_{C}$ of a cube $C\in Q$ in a similar way; namely (cf. ((3.2))),

1. (3.3)

if $C=(X\,|\,T)$ and $T=(p_{1}<\ldots<p_{d}))$, then $t_{C}=X\cup\{p_{i}\colon d-i$ odd$\}$ and $h_{C}=X\cup\{p_{i}\colon d-i$ even$\}$.

Also for each vertex $v$ of $Q$, unless $v$ is in $Z^{\,\rm rear}$, there is a unique cube $C\in Q$ such that $t_{C}=v$, and symmetrically, unless $v$ is in $Z^{\,\rm fr}$, there is a unique cube $C\in Q$ such that $h_{C}=v$ (to see this, consider the line going through $v$ and parallel to $e_{d}$).

Therefore, by drawing for each cube $C\in Q$, the edge-arrow from $t_{C}$ to $h_{C}$, we obtain a directed graph whose connectivity components are directed paths beginning at $Z^{\,\rm fr}-Z^{\,\rm rim}$ and ending at $Z^{\,\rm rear}-Z^{\,\rm rim}$. We call these paths bead-threads in $Q$. It is convenient to add to this graph the elements of $V(Z^{\,\rm rim})$ as isolated vertices, forming degenerate bead-threads, each going from a vertex to itself. Let $B_{Q}$ be the resulting directed graph. Then

1. (3.4)

$B_{Q}$ contains all vertices of $Q$, and each component of $B_{Q}$ is a bead-thread going from $Z^{\,\rm fr}$ to $Z^{\,\rm rear}$.

Note that the heights $|X|$ of vertices $X$ along a bead-thread are monotone increasing when $d$ is odd, and constant when $d$ is even.

4 Proof of Theorem 1.1

Let $r$ be odd and $n>r$. We have to show that

1. (4.1)

if ${\cal W}$ is a weakly $r$-separated collection in $2^{[n]}$, then $|{\cal W}|\leq\binom{n}{\leq r+1}$.

This is valid when $r=1$ (cf. ((1.1))) and is trivial when $n=r+1$. So one may assume that $3\leq r\leq n-2$. We prove ((4.1)) by induction, assuming that the corresponding inequality holds for ${\cal W}^{\prime},n^{\prime},r^{\prime}$ when $n^{\prime}\leq n$, $r^{\prime}\leq r$, and $(n^{\prime},r^{\prime})\neq(n,r)$.

Define the following subcollections in ${\cal W}$:

[TABLE]

referring to elements $A,An$ in ${\cal W}$ as twins. Observe that

1. (4.2)

any $A,B\in{\cal W}^{-}$ are weakly $r$-separated.

Indeed, this is trivial when $A,B\in{\cal W}$ or $An,Bn\in{\cal W}$. Assume that $A\in{\cal W}$ and $B^{\prime}:=Bn\in{\cal W}$, and that $A,B^{\prime}$ are $(r+2)$-interlaced (for if $A,B^{\prime}$ are $r^{\prime}$-interlaced with $r^{\prime}\leq r+1$, then so is for $A,B$, and we are done). Since $\max(B^{\prime}-A)=n>\max(A-B^{\prime})$ and $r+2$ is odd, $B^{\prime}$ surrounds $A$. Therefore, $\min(B^{\prime}-A)<\min(A-B^{\prime})$ and $|B^{\prime}|\leq|A|$. Then $|B|<|A|$ and $\min(B-A)=\min(B^{\prime}-A)<\min(A-B)$, implying that $A,B$ are weakly $r$-separated, as required.

By induction, $|{\cal W}^{-}|\leq\binom{n-1}{\leq r+1}$. Also one can observe that $|{\cal W}|=|{\cal W}^{-}|+|{\cal T}|$. Therefore, using the identity $\binom{n}{j}=\binom{n-1}{j}+\binom{n-1}{j-1}$ for any $j\leq n-1$, in order to obtain the inequality in ((4.1)), it suffices to show that

[TABLE]

For $i=0,1,\ldots n-1$, define ${\cal T}^{i}:=\{A\in{\cal T}\,\colon\,|A|=i\}$. We will rely on two claims.

Claim 1  For each $i$, the collection ${\cal T}^{i}$ is $(r-1)$-separated; moreover, ${\cal T}^{i}$ is weakly $(r-2)$-separated.

Proof  Let $A,B\in{\cal T}^{i}$. Take the interval cortege $(I_{1},\ldots,I_{r^{\prime}})$ for $A,B$, and let for definiteness $I_{r^{\prime}}\subseteq A-B$. Then $(I_{1},\ldots,I_{r^{\prime}},I_{r^{\prime}+1}:=\{n\})$ is the interval cortege for $A$ and $B^{\prime}:=Bn$. Since $|A|<|B^{\prime}|$ and $\max(A-B^{\prime})<\max(B^{\prime}-A)=n$, the fact that $A,B^{\prime}$ are weakly $r$-separated implies that $r^{\prime}+1$ is strictly less than $r+2$. Then $r^{\prime}\leq r$, which means that $A,B$ are $(r-1)$-separated. Since $|A|=|B|$ and $r$ is odd, we also can conclude that $A,B$ are weakly $(r-2)$-separated.

Now consider the zonotope $Z=Z(n-1,r)$. For $j=0,1,\ldots,n-1$, define ${\cal S}^{j}$ (${\cal A}^{j}$) to be the set of vertices $X$ of $Z^{\,\rm fr}$ (resp. $Z^{\,\rm rear}$) with $|X|=j$. We extend each collection ${\cal T}^{i}$ to ${\cal D}^{i}$, defined as

[TABLE]

Claim 2  ${\cal D}^{i}$ is weakly $(r-2)$-separated.

Proof  The vertex sets of $Z^{\,\rm fr}$ and $\pi(Z^{\,\rm fr})$ are essentially the same (regarding a vertex as a subset of $[n-1]$), and similarly for $Z^{\,\rm rear}$ and $\pi(Z^{\,\rm rear})$. Since $\pi(Z^{\,\rm fr})$ and $\pi(Z^{\,\rm rear})$ are cubillages on $Z(n-1,r-1)$ (the so-called “standard” and “anti-standard” ones),  ((2.2)) implies that both collections $V(Z^{\,\rm fr})={\cal S}^{0}\cup\ldots\cup{\cal S}^{n-1}$ and $V(Z^{\,\rm rear})={\cal A}^{0}\cup\ldots\cup{\cal A}^{n-1}$ are $(r-2)$-separated, and therefore, they are weakly $(r-2)$-separated as well.

Next, by ((2.3))(i), each vertex $X$ of $Z^{\,\rm fr}$ is a $k$-interval with $k\leq(r-1)/2$. Such an $X$ and any subset $Y\subseteq[n-1]$ are $k^{\prime}$-interlaced with $k^{\prime}\leq 2k+1$. Then $k^{\prime}\leq r$ and this holds with equality when $X$ and $Y$ are $r$-interlaced and $Y$ surrounds $X$. It follows that $X$ is weakly $(r-2)$-separated from any $Y\subseteq[n-1]$ with $|Y|\leq|X|$ (in particular, if $X\in{\cal S}^{j}$ and $j\geq i$, then $X$ is weakly $(r-2)$-separated from each member of ${\cal T}^{i}\cup{\cal A}^{0}\cup\ldots\cup{\cal A}^{i-1}$).

Symmetrically, by ((2.3))(ii), each vertex $X$ of $Z^{\,\rm rear}$ is the complement (to $[n-1]$) of a $k$-interval with $k\leq(r-1)/2$. We can conclude that such an $X$ is weakly $(r-2)$-separated from any $Y\subseteq[n-1]$ with $|Y|\geq|X|$.

Now the result is provided by the inequalities $|X|>|A|>|X^{\prime}|$ for any $X\in{\cal S}^{i+1}\cup\ldots\cup{\cal S}^{n-1}$,  $A\in{\cal T}^{i}$, and $X^{\prime}\in{\cal A}^{0}\cup\ldots\cup{\cal A}^{i-1}$.

By induction, $|{\cal D}^{i}|\leq\binom{n-1}{\leq r-1}$. Then, using ((1.2)) and ((3.1)) (relative to $n-1$ and $r-2$), we have

[TABLE]

Let ${\cal S}^{\prime}:={\cal S}^{0}\cup{\cal S}^{1}\cup\ldots\cup{\cal S}^{i}$ and ${\cal A}^{\prime}:={\cal A}^{0}\cup{\cal A}^{1}\cup\ldots\cup{\cal A}^{i-1}$. Since ${\cal S}^{i+1}\cup\ldots\cup{\cal S}^{n-1}=V(Z^{\,\rm fr})-{\cal S}^{\prime}$, we obtain from (4.4) and (4.5) that

[TABLE]

We now finish the proof by using a bead-thread techniques (as in Sect. 3). Fix an arbitrary cubillage $Q$ in $Z=Z(n-1,r)$. Let ${\cal R}^{i}$ be the set of vertices $X$ of $Q$ with $|X|=i$, and let ${\cal B}$ be the set of paths (bead-threads) in the graph $B_{Q}$ beginning at $Z^{\,\rm fr}$ and ending at $Z^{\,\rm rear}$. Since $r$ is odd, each edge $(X,Y)$ of $B_{Q}$ is “ascending” (satisfies $|Y|>|X|$). This implies that each path $P\in{\cal P}$ beginning at ${\cal S}^{\prime}$ must meet either ${\cal R}^{i}$ or ${\cal A}^{\prime}$, and conversely, each path meeting ${\cal R}^{i}\cup{\cal A}^{\prime}$ begins at ${\cal S}^{\prime}$. This and (4.6) imply

[TABLE]

Summing up these inequalities for $i=0,1,\ldots,n-1$, we have

[TABLE]

yielding (4.3) and completing the proof of Theorem 1.1.

5 Proof of Theorem 1.2

Let $r,\,r^{\prime},\,P=\{p_{1},\ldots,p_{r^{\prime}}\},\,Q=\{q_{0},\ldots,q_{r^{\prime}}\}$ and $X$ be as in the hypotheses of Theorem 1.2 (where $r$ is odd and $r^{\prime}=(r+1)/2$). We will use the following notation and terminology.

Let $A,B\subset[n]$. The interval cortege for $A,B$ is denoted by ${\cal I}(A,B)$, and when it is not confusing, we refer to the intervals in it concerning $A-B$ ($B-A$) as $A$-bricks (resp. $B$-bricks). When $A\cap B=\emptyset$, we may abbreviate $A\cup B$ as $AB$. When $A,B$ are not weakly $r$-separated, we say that the pair $\{A,B\}$ is bad.

Note that for $P,Q,X$ as above and for the set ${\cal N}(P,Q)$ of neighbors of $P,Q$ (defined in (1.4)), there hold: $PX$ and $QX$ are $(r+2)$-interlaced; $XQ$ surrounds $XP$;  $|XQ|>|XP|$; and $\{XP,XQ\}$ is the unique bad pair in the collection $\{XS\colon S\in\{P,Q\}\cup{\cal N}(P,Q)\}$.

We are going to obtain a sharper version of Theorem 1.2. In particular, it deals with only $O(r^{2})$ (rather than exponentially many) neighbors of $(P,Q)$.

Theorem 5.1

Let $r,n,P,Q,X$ be as above. Define

[TABLE]

Let $Y\subset[n]$ be different from $XP$ and $XQ$. Then:

(i)* if $\{Y,XP\}$ is bad, then there exists $S\in\ {\cal N}^{\uparrow}(P,Q)$ such that $\{Y,XS\}$ is bad;*

(ii)* if $\{Y,XQ\}$ is bad, then there exists $S\in\ {\cal N}^{\downarrow}(P,Q)$ such that $\{Y,XS\}$ is bad.*

(To obtain Theorem 1.2, consider ${\cal W},U$ as in that theorem and let $U^{\prime}$ be the member of $\{XP,XQ\}$ different from $U$. Suppose that $\{Y,U^{\prime}\}$ is bad for some $Y\in{\cal W}-\{U\}$. By Theorem 5.1 applied to $Y,U^{\prime}$, there exists $S\in{\cal N}(P,Q)$ such that $\{Y,XS\}$ is bad. But ${\cal W}$ is weakly $r$-separated and contains both $Y$ and $XS$.)

Proof  W.l.o.g., one may assume that $Y\cap X=\emptyset$. We first prove assertion (i) (obtaining (ii) as a consequence, as we explain in the end of the proof). We will abbreviate the neighbor set ${\cal N}^{\uparrow}(P,Q)$ as ${\cal N}^{\uparrow}$, and the interval cortege ${\cal I}(Y,XP)$ as ${\cal I}$.

Suppose, for a contradiction, that no pair $\{Y,XS\}$ with $S\in{\cal N}^{\uparrow}$ is bad. This will impose restrictions on $Y$ and eventually will lead us to the conclusion that $Y$ is impossible.

The core of the proof consists in the next lemma. Here we refer to an element $p\in P$ ($q\in Q$) as refined if it forms the single-element $XP$-brick $\{p\}$ (resp. the single-element $Y$-brick $\{q\}$) in ${\cal I}$.

Lemma 5.2

Let $Y\subset[n]$ be different from $XP,XQ$ and suppose that for each $S\in{\cal N}^{\uparrow}(P,Q)$, the sets $Y$ and $XS$ are weakly $r$-separated (while $\{Y,XP\}$ is bad). Then at least one of the following holds:

($\ast$)*  all elements of $P$ are refined;*

($\ast\ast$)*  all elements of $Q$ are refined.*

This lemma will be proved later, and now assuming its validity, we finish the proof of the theorem as follows. Note that $Y\cap P\neq\emptyset$ is possible (whereas $Y\cap X=\emptyset$, as is assumed above).

Let $a$ and $b$ denote the numbers of $Y$- and $XP$-bricks in ${\cal I}$, respectively. Then $a+b=|{\cal I}|\geq r+2=2r^{\prime}+1$ and $|a-b|\leq 1$. We assume that the intervals in ${\cal I}$ are viewed as $\ldots<A_{i-1}<B_{i}<A_{i}<B_{i+1}\ldots$ , where $A_{i^{\prime}}$ ($B_{i^{\prime}}$) stands for a $Y$-brick (resp. $XP$-brick). The first (last) $Y$-brick is denoted by $A^{m}$ (resp. $A^{M}$), and the first (last) $XP$-brick by $B^{m}$ (resp. $B^{M}$). Also for a set $C\subset[n]$ and a singleton $c\in[n]$, we write $c<C$ ($c>C$) if $c<\min(C)$ (resp. $c>\max(C)$).

We first assume that $(\ast\ast)$ from Lemma 5.2 is valid. Then $a\geq|Q|=r^{\prime}+1$. Consider two possible cases for $a$.

Case I: $a\geq r^{\prime}+2$. Then $b\geq r^{\prime}+1$ and $|{\cal I}|\geq 2r^{\prime}+3$. If $q_{0}<B^{m}$, then, obviously, $A^{m}=\{q_{0}\}$. Taking $S:=Pq_{0}\in{\cal N}^{\uparrow}(P,Q)$, we obtain $|{\cal I}(Y,XS)|=|{\cal I}|-1\geq 2r^{\prime}+2$ (since the $Y$-brick $\{q_{0}\}$ disappears, while the other bricks of ${\cal I}$ preserve). Hence $\{Y,XS\}$ is bad. Similarly, if $B^{M}<q_{r^{\prime}}$, then $A^{M}=\{q_{r^{\prime}}\}$, and taking $S:=Pq_{r^{\prime}}$, we again obtain $|{\cal I}(Y,XS)|\geq 2r^{\prime}+2$, whence $\{Y,XS\}$ is bad.

So we may assume that $B^{m}<q_{0}$ and $q_{r^{\prime}}<B^{M}$. Then $b\geq r^{\prime}+2$ and $|{\cal I}|=a+b\geq 2r^{\prime}+4$. Taking $S:=Pq_{0}$, we obtain $|{\cal I}(Y,XS)|=|{\cal I}|-2\geq 2r^{\prime}+2$ (since the $Y$-brick $\{q_{0}\}$ disappears and the $X..$-bricks preceding and succeeding $\{q_{0}\}$ merge). Thus, in all cases, $\{Y,XS\}$ is bad; a contradiction.

Case II: $a=r^{\prime}+1$. Then $A^{m}=\{q_{0}\},\{q_{1}\},\ldots,\{q_{r^{\prime}}\}=A^{M}$ are exactly the $Y$-bricks of ${\cal I}$. If $B^{m}<q_{0}$ and $B^{M}<q_{r^{\prime}}$, then $b=a=r^{\prime}+1$ and $|{\cal I}|=2r^{\prime}+2$. Taking $S:=Pq_{r^{\prime}}$, we obtain $|{\cal I}(Y,XS)|=|{\cal I}|-1=2r^{\prime}+1$. Also $XS$ surrounds $Y$ (since $B^{M}$ becomes the last interval in ${\cal I}(Y,XS)$). Hence $|Y-XS|=r^{\prime}<r^{\prime}+1\leq|XS-Y|$, implying that $\{Y,XS\}$ is bad.

Similarly, if $q_{0}<B^{m}$ and $q_{r^{\prime}}<B^{M}$, then $S:=Pq_{0}$ gives $|{\cal I}(Y,XS)|=|{\cal I}|-1=2r^{\prime}+1$, and $XS$ surrounds $Y$ as well. And if $B^{m}<q_{0}$ and $q_{r^{\prime}}<B^{M}$, then $b=r^{\prime}+2$, and for $S:=Pq_{0}$, we obtain $|{\cal I}(Y,XS)|=|{\cal I}|-2=2r^{\prime}+1$. Again, $XS$ surrounds $Y$, whence $\{Y,XS\}$ is bad.

So it remains to consider the situation when $q_{0}<B^{m}$ and $B^{M}<q_{r^{\prime}}$. Then $b=r^{\prime}$ and $Y$ surrounds $XP$. Since $\{Y,XP\}$ is bad and $Y-XP=Q$, we have $r^{\prime}+1=|Y-XP|>|XP-Y|\geq r^{\prime}$. It follows that $|XP-Y|=r^{\prime}$. This implies that each $XP$-brick is a singleton, and that if $Y\cap P=\emptyset$, then the $XP$-bricks of ${\cal I}$ are exactly $\{p_{1}\},\ldots,\{p_{r^{\prime}}\}$. But then $X=\emptyset$ and $Y=Q=XQ$, contradicting the hypotheses of the theorem.

Therefore, $Y$ must contain an element $p_{i}$ for some $i$. Then $p_{i}\notin B_{i}$ (in view of $|B_{i}|=1$ and $p_{i}\notin XP-Y$). So one of two situations takes place: $q_{i-1}<p_{i}<B_{i}<q_{i}$, or $q_{i-1}<B_{i}<p_{i}<q_{i}$. Define $S:=(P-p_{i})q_{i-1}$ in the former case, and $S:=(P-p_{i})q_{i}$ in the latter case. The transformation $XP\mapsto XS$ replaces the $Y$-brick $\{q_{i-1}\}$ or $\{q_{i}\}$ by $\{p_{i}\}$. We have: $|{\cal I}(Y,XS)|=|{\cal I}|=2r^{\prime}+1$, $Y$ surrounds $XS$, and $|Y|>|XS|$. Hence $\{Y,XS\}$ is bad; a contradiction.

Next we assume that $(\ast)$ from Lemma 5.2 is valid. Then $b\geq r^{\prime}$ and each $p_{i}\in P$ forms the $XP$-brick $\{p_{i}\}$ in ${\cal I}$ (admitting other $XP$-bricks); in particular, $Y\cap P=\emptyset$. Consider two possibilities for $b$.

Case III: $b=r^{\prime}$. Then $\{p_{1}\},\ldots,\{p_{r^{\prime}}\}$ are exactly the $XP$-bricks of ${\cal I}$, $X=\emptyset$, and $a=r^{\prime}+1$ (in view of $|{\cal I}|\geq 2r^{\prime}+1$). So $Y$ surrounds $P=XP$, and $|Y|>|P|$. Assuming that $(\ast\ast)$ from Lemma 5.2 is violated, there is $i\in\{0,\ldots,r^{\prime}\}$ such that $\{q_{i}\}$ is not a $Y$-brick of ${\cal I}$. Then either (a) $q_{i}$ lies in some $Y$-brick $A_{j}$ with $|A_{j}|\geq 2$, or (b) $q_{i}$ is in none of the intervals of ${\cal I}$.

In case (a), set $S:=Pq_{i}$. Note that $|Y|\geq r^{\prime}+2=|P|+2$ (in view of $a=r^{\prime}+1$ and $|A_{j}|\geq 2$). If $q_{i}\in Y$, then the transformation $P\mapsto S$ replaces $A_{j}$ by a (possibly smaller but nonempty) $Y$-brick in ${\cal I}(Y,S)$, while preserving the other intervals of ${\cal I}$. It follows that $|{\cal I}(Y,S)|=|{\cal I}|=2r^{\prime}+1$, $Y$ surrounds $S$, and $|Y|>|P|+1=|S|$; so $\{Y,S\}$ is bad. And if $q_{i}\notin Y$, then, obviously, $\min(A_{j})<q_{i}<\max(A_{j})$. This gives $|{\cal I}(Y,S)|=|{\cal I}|+2$ (since $P\mapsto S$ replaces $A_{j}$ by the $S$-brick $\{q_{i}\}$ and two $Y$-bricks, one containing $\min(A_{j})$ and the other containing $\max(A_{j})$); so $\{Y,S\}$ is bad again.

And in case (b), obviously, $q_{i}\notin Y$. Then one of four subcases takes place: (b1) $p_{i}<q_{i}<A_{i}$; (b2) $A_{i}<q_{i}<p_{i+1}$; (b3) $i=0$ and $q_{0}<A^{m}$; and (b4) $i=r^{\prime}$ and $A^{M}<q_{r^{\prime}}$. In subcases (b3) and (b4), for $S:=Pq_{i}$, we have $|{\cal I}(Y,S)|=|{\cal I}|+1$ (since $\{q_{i}\}$ becomes a new brick), whence $\{Y,S\}$ is bad. In subcases (b1) and (b2), for $S^{\prime}:=(P-p_{i})q_{i}$ and $S^{\prime}:=(P-p_{i+1})q_{i}$, respectively, the transformation $P\mapsto S^{\prime}$ replaces the $P$-brick $\{p_{i}\}$ or $\{p_{i+1}\}$ by the $S^{\prime}$-brick $\{q_{i}\}$, and the badness of $\{Y,P\}$ implies that of $\{Y,S^{\prime}\}$.

Case IV: $b\geq r^{\prime}+1$. Assuming, as before, that we are not in $(\ast\ast)$ from Lemma 5.2, there is $i$ such that $\{q_{i}\}$ is not a $Y$-brick of ${\cal I}$. Take $S:=Pq_{i}$. We can observe that in all possible cases for $q_{i}$ (exposed in Case III above), the transformation $XP\mapsto XS$ leads to the following: $|XS|>|XP|$, each $Y$-brick of ${\cal I}$ either preserves or is replaced by a (nonempty) $Y$-brick of ${\cal I}(Y,XS)=:{\cal I}^{\prime}$, and similarly for the $XP$-bricks of ${\cal I}$. Then $|{\cal I}^{\prime}|\geq|{\cal I}|\geq 2r^{\prime}+1$ and, moreover, in case $|{\cal I}^{\prime}|=2r^{\prime}+1$, the number of $XS$-bricks ($Y$-bricks) of ${\cal I}^{\prime}$ is the same as the one of $XP$-bricks (resp. $Y$-bricks) of ${\cal I}$, which is equal to $r^{\prime}+1=b$ (resp. $r^{\prime}=a$). So $b>a$, $XP$ surrounds $Y$, and $|XP|>|Y|$ (since $\{Y,XP\}$ is bad). Now the badness of $\{Y,XS\}$ follows from $|XS|>|XP|$, yielding a contradiction.

Thus, assertion (i) in the theorem is proven.

It remains to show (ii). We reduce it to the previous case, using the following observation. For $A\subseteq[n]$, let $\overline{A}$ denote the complementary set $[n]-A$. One can see that ${\cal I}(A,B)={\cal I}(\overline{A},\overline{B})$ and that the bricks for $A-B$ coincide with those for $\overline{B}-\overline{A}$. It follows that if $A,B$ are $(r+2)$-interlaced (where $r$ is odd, as before) and $A$ surrounds $B$, then $\overline{A},\overline{B}$ are $(r+2)$-interlaced, $\overline{B}$ surrounds $\overline{A}$, and $|A|-|B|=|\overline{B}|-|\overline{A}|$. Therefore, if $A,B$ are weakly $r$-separated then so are $\overline{A},\overline{B}$.

Now for $Y,P,Q,X$ as above and $U:=XQ$, consider $Y^{\prime}:=\overline{Y}$,  $X^{\prime}:=\overline{XPQ}$ ($=[n]-(X\cup P\cup Q)$) and $U^{\prime}:=\overline{XQ}$. Suppose that $\{Y,U\}$ is bad. Then $\{Y^{\prime},U^{\prime}\}$ is bad as well. Note also that $U^{\prime}=X^{\prime}P$. By the theorem applied to $Y^{\prime},P,Q,X^{\prime}$ and $U^{\prime}$, there exists $S^{\prime}\in{\cal N}^{\uparrow}(P,Q)$ such that $\{Y^{\prime},X^{\prime}S^{\prime}\}$ is bad. Then $\{Y,\overline{X^{\prime}S^{\prime}}\}$ is bad as well. Take $S:=(P\cup Q)-S^{\prime}$. One can see that $S\in{\cal N}^{\downarrow}(P,Q)$ and $\overline{X^{\prime}S^{\prime}}=XS$. Therefore, $\{Y,XS\}$ is bad, as required.

This completes the proof of Theorem 5.1 (implying Theorem 1.2).

Note that Theorem 5.1 implies a sharper version of the theorem on combinatorial flips.

Corollary 5.3

For $r,n,P,Q,X$ as in Theorem 1.2, if a weakly $r$-separated collection ${\cal W}\subset 2^{[n]}$ contains the set $XP$ ($XQ$) and the sets $XS$ for all $S\in\ {\cal N}^{\downarrow}(P,Q)$ (resp. $S\in\ {\cal N}^{\uparrow}(P,Q)$), then the collection obtained from ${\cal W}$ by replacing $XP$ by $XQ$ (resp. $XQ$ by $XP$) is weakly $r$-separated as well.

We finish this section with proving the above-mentioned lemma.

Proof of Lemma 5.2  Suppose that there are simultaneously $p\in P$ and $q\in Q$ that are not refined. Form $S^{\prime}:=P-p$, $S^{\prime\prime}:=Pq$ and $S:=(P-p)q$ (note that $S^{\prime\prime}$ and $S$ are in ${\cal N}^{\uparrow}={\cal N}^{\uparrow}(P,Q)$, whereas $S^{\prime}$ is not). Let ${\cal I}:={\cal I}(Y,XP)$, ${\cal I}^{\prime}:={\cal I}(Y,XS^{\prime})$ and ${\cal I}^{\prime\prime}:={\cal I}(Y,XS^{\prime\prime})$. We write $A_{i}$ ($B_{i}$) for $Y$-bricks (resp. $XP$-bricks) in ${\cal I}$ and assume that they follow in ${\cal I}$ in the order $\ldots<A_{i-1}<B_{i}<A_{i}<B_{i+1}\ldots$ .

For $A,B,A^{\prime},B^{\prime}\subseteq[n]$, let us say that the ordered pairs $(A,B)$ and $(A^{\prime},B^{\prime})$ have the same type if $|{\cal I}(A,B)|=|{\cal I}(A^{\prime},B^{\prime})|$ and the first interval of ${\cal I}(A,B)$ concerns $A-B$ if and only if the first interval of ${\cal I}(A^{\prime},B^{\prime})$ concerns $A^{\prime}-B^{\prime}$ (implying that a similar property holds for the last intervals of ${\cal I}(A,B)$ and ${\cal I}^{\prime}(A^{\prime},B^{\prime})$).

We examine four possible cases for $p,q$.

Case 1:  $p$ lies in an interval $C$ of ${\cal I}$.

(1a)  Suppose that $C$ is an $XP$-brick $B_{i}$. Since $p$ is not refined, $|B_{i}|\geq 2$. If $p\notin Y$, then the transformation $XP\mapsto XS^{\prime}$ replaces $B_{i}$ by a (nonempty) $XS^{\prime}$-brick $B^{\prime}_{i}$ in ${\cal I}$ with $B^{\prime}_{i}\subseteq B_{i}$, while the other intervals in ${\cal I}$ and ${\cal I}^{\prime}$ coincide. Therefore, ${\cal I}^{\prime}$ and ${\cal I}$ have the same type.

And if $p\in Y$, then $\min(B_{i})<p<\max(B_{i})$, and $XP\mapsto XS^{\prime}$ replaces $B_{i}$ by three bricks, say, $B^{\prime}<A^{\prime}<B^{\prime\prime}$, where $A^{\prime}$ is the single-element $Y$-brick $\{p\}$, and $B^{\prime},B^{\prime\prime}$ are $XS^{\prime}$-bricks (with $\min(B^{\prime})=\min(B_{i})$ and $\max(B^{\prime\prime})=\max(B_{i})$). Then $|{\cal I}^{\prime}|=|{\cal I}|+2$.

(1b)  Now suppose that $C$ is a $Y$-brick $A_{i}$. This is possible only if $p\in Y$ and $\min(A_{i})<p<\max(A_{i})$. Then $p\in Y-XS^{\prime}$, and $XP\mapsto XS^{\prime}$ preserves $A_{i}$ (as well as the other intervals of ${\cal I}$), whence ${\cal I}^{\prime}={\cal I}$.

Case 2:  $q$ lies in an interval $C$ of ${\cal I}$.

(2a) Suppose that $C=B_{i}$. This is possible only if $q\notin Y$ and $\min(B_{i})<q<\max(B_{i})$ (since $q\notin XP$). Then $XP\mapsto XS^{\prime\prime}$ preserves $B_{i}$, yielding ${\cal I}^{\prime\prime}={\cal I}$.

(2b) Now suppose that $C=A_{i}$. Since $q$ is not refined, $|A_{i}|\geq 2$. If $q\in Y$, then $XP\mapsto XS^{\prime\prime}$ replaces $A_{i}$ by a (nonempty) $Y$-brick $A^{\prime}_{i}$ with $A^{\prime}_{i}\subseteq A_{i}$. Therefore, ${\cal I}^{\prime\prime}$ and ${\cal I}$ have the same type. And if $q\notin Y$, then $XP\mapsto XS^{\prime\prime}$ replaces $A_{i}$ by three bricks $A^{\prime}<B^{\prime}<A^{\prime\prime}$, where $B^{\prime}$ is the $XS^{\prime\prime}$-brick $\{q\}$, and $A^{\prime},A^{\prime\prime}$ are $Y$-bricks (with $\min(A^{\prime})=\min(A_{i})$ and $\max(A^{\prime\prime})=\max(A_{i})$), whence $|{\cal I}^{\prime\prime}|=|{\cal I}|+2$.

Case 3:  $p$ belongs to no interval of ${\cal I}$. Then $p\in Y$.

(3a) Suppose that $A_{i}<p<B_{i+1}$ or $B_{i}<p<A_{i}$ for some $i$. Then $p\in Y-XS^{\prime}$, and $XP\mapsto XS^{\prime}$ extends $A_{i}$ (making a $Y$-brick with the beginning or end at $p$). Hence ${\cal I}^{\prime}$ and ${\cal I}$ have the same type.

(3b) Suppose that $p<C$, where $C$ is the first interval of ${\cal I}$. If $C$ is an $XP$-brick, then $XP\mapsto XS^{\prime}$ produces a new $Y$-brick, namely, $\{p\}$, and preserves the other intervals of ${\cal I}$, whence $|{\cal I}^{\prime}|=|{\cal I}|+1$. And if $C$ is a $Y$-brick, then $XP\mapsto XS^{\prime}$ extends $C$ (making a $Y$-brick with the beginning $p$), whence ${\cal I}^{\prime}$ and ${\cal I}$ have the same type.

(3c) Similarly, if $p>D$, where $D$ is the last interval of ${\cal I}$, then either $|{\cal I}^{\prime}|=|{\cal I}|+1$, or ${\cal I}^{\prime}$ and ${\cal I}$ have the same type (when $D$ is extended to a $Y$-brick with the end $p$).

Case 4:  $q$ belongs to no interval of ${\cal I}$. Then $q\notin Y$.

(4a) Suppose that $A_{i-1}<q<B_{i}$ or $B_{i}<q<A_{i}$ for some $i$. Then $XP\mapsto XS^{\prime\prime}$ extends $B_{i}$ (making a $XS^{\prime\prime}$-brick with the beginning or end at $q$). Hence ${\cal I}^{\prime\prime},{\cal I}$ have the same type.

(4b) Suppose that $q<C$, where $C$ is the first interval of ${\cal I}$. If $C$ is an $XP$-brick, then $XP\mapsto XS^{\prime\prime}$ extends $C$ (making an $XS^{\prime\prime}$-brick with the beginning $q$), whence ${\cal I}^{\prime\prime},{\cal I}$ have the same type. And if $C$ is a $Y$-brick, then $XP\mapsto XS^{\prime\prime}$ preserves $C$ and produces a new $X..$-brick $\{q\}$, whence $|{\cal I}^{\prime\prime}|=|{\cal I}|+1$.

(4c) Similarly, if $q>D$, where $D$ is the last interval of ${\cal I}$, then either ${\cal I}^{\prime\prime}$ and ${\cal I}$ have the same type, or $|{\cal I}^{\prime\prime}|=|{\cal I}|+1$.

Now we finish proving the lemma as follows. Observe that in the above cases, $|{\cal I}^{\prime}|\geq|{\cal I}|$ is valid throughout, and if this holds with equality, then ${\cal I}^{\prime}$ and ${\cal I}$ have the same type. For ${\cal I}^{\prime\prime}$ and ${\cal I}$, the behavior is similar.

If $|{\cal I}^{\prime\prime}|>|{\cal I}|$ happens, then $Y$ and $XS^{\prime\prime}$ form a bad pair (since they are $|{\cal I}^{\prime\prime}|$-interlaced with $|{\cal I}^{\prime\prime}|>r+2$ and taking into account that $S^{\prime\prime}=Pq\in{\cal N}^{\uparrow}$). This contradicts the hypotheses of the lemma.

Now let $|{\cal I}^{\prime\prime}|=|{\cal I}|$ (and ${\cal I}^{\prime\prime},{\cal I}$ have the same type). Then we consider the neighbor $S=(P-p)q\in{\cal N}^{\uparrow}$ and assert that $\{Y,XS\}$ is bad, thus coming to a contradiction again.

To show this, let $\widetilde{\cal I}:={\cal I}(Y,XS)$. Suppose that $q\in Y$. Forming $Y^{-}:=Y-q$, we have $Y^{-}-XP=Y-XS^{\prime\prime}$ and $XP-Y^{-}=XS^{\prime\prime}-Y$, implying that ${\cal I}^{-}:={\cal I}(Y^{-},XP)$ coincides with ${\cal I}^{\prime\prime}$. Hence ${\cal I}^{-}$ and ${\cal I}$ have the same type. Moreover, under the correspondence of intervals in these corteges (exposed in (2b)), each $Y^{-}$-brick of ${\cal I}^{-}$ is included in the corresponding $Y$-brick of ${\cal I}$, and each $XP$-brick of ${\cal I}^{-}$ includes the corresponding $XP$-brick of ${\cal I}$. In particular, $p$ is not refined w.r.t. ${\cal I}^{-}$. So we can apply to $X,P,Y^{-},p$ reasonings as in Cases 1 and 3 and conclude that under the transformation $XP\mapsto XS^{\prime}$, the cortege ${\cal I}^{-}$ turns into $\widehat{\cal I}:={\cal I}(Y^{-},XS^{\prime})$ such that either $|\widehat{\cal I}|>|{\cal I}^{-}|$, or $\widehat{\cal I}$ and ${\cal I}^{-}$ have the same type. But $Y=Y^{-}q$ and $S=S^{\prime}q$ imply $\widehat{\cal I}=\widetilde{\cal I}$. Now the badness of $\{Y,XS\}$ is immediate when $|\widehat{\cal I}|>|{\cal I}^{-}|$ ($=|{\cal I}|$), and follows from the badness of $\{Y,XP\}$ when $|\widehat{\cal I}|=|{\cal I}^{-}|$ (since $\widehat{\cal I}$ and ${\cal I}$ have the same type and $|Y^{-}|-|XS^{\prime}|=|Y|-|XS|=|Y|-|XP|$).

Finally, let $q\notin Y$. Then (in view of $|{\cal I}^{\prime\prime}|=|{\cal I}|$) we are in one of the following subcases: (2a) with $\min(B_{i})<q<\max(B_{i})$ for some $i$; or (4a) with $A_{i-1}<q<B_{i}$ or $B_{i}<q<A_{i}$ for some $i$; or (4b) with $q<B^{m}<A^{m}$; or (4c) with $q>B^{M}>A^{M}$ (where $A^{m}$ and $A^{M}$ (resp. $B^{m}$ and $B^{M}$) are the first and last $Y$-bricks (resp. $XP$-bricks) in ${\cal I}$, respectively). By explanations above, in all of these situations, $XP\mapsto XS^{\prime\prime}$ leads to increasing at most one of $X..$-bricks and preserving the other intervals of ${\cal I}$. This implies that $p$ is not refined w.r.t. ${\cal I}^{\prime\prime}$, and we can apply to $X,S^{\prime\prime},Y,p$ reasonings as in Cases 1 and 3 and conclude that $XS^{\prime\prime}\mapsto XS$ turns ${\cal I}^{\prime\prime}$ into $\widetilde{\cal I}$ so that either $|\widetilde{\cal I}|>|{\cal I}^{\prime\prime}|$ ($=|{\cal I}|$), or $\widetilde{\cal I}$ and ${\cal I}^{\prime\prime}$ have the same type. Then the badness of $\{Y,XS\}$ follows.

This completes the proof of the lemma.

6 Weakly $r$-separated collections generated by cubillages

In Sects. 2, 3 we outlined an interrelation between (strongly) $\ast$-separated collections on one hand, and cubillages and s-membranes on other hand (see ((2.2)) and ((3.1))). This section is devoted to geometric aspects of the weak $r$-separation, assuming that $r$ is odd. Being motivated by geometric constructions for maximal weakly 1-separated collections elaborated in [3, 4], we explain how to construct maximal by size weakly $r$-separated collections by use of the so-called w-membranes; these are analogs of s-membranes in certain fragmentations of cubillages.

In subsections below we introduce the notions of fragmentation and w-membrane, demonstrate their properties (extending results from [4, Sect. 6]) and finish with a theorem saying that the vertex set of any $(r+1)$-dimensional w-membrane gives rise to a maximal by size weakly $r$-separated collection (for corresponding $n$). Note that in Sects. 6.1–6.3 the dimension $d$ of a zonotope/cubillage in question is assumed to be arbitrary (not necessarily odd).

6.1 Fragmentation

Let $Q$ be a cubillage on $Z(n,d)$. For $\ell=0,1,\ldots,n$, we denote the “horizontal” hyperplane at “height” $\ell$ in ${\mathbb{R}}^{d}$ by $H_{\ell}$, i.e., $H_{\ell}:=\{x=(x_{1},\ldots,x_{d})\in{\mathbb{R}}^{d}\,\colon\,x_{1}=\ell\}$. The fragmentation of $Q$ is meant to be the complex $Q^{\equiv}$ obtained by cutting $Q$ by $H_{1},\ldots,H_{n-1}$.

Such hyperplanes subdivide each cube $C=(X\,|\,T)$ of $Q$ into $d$ pieces $C^{\equiv}_{1},\ldots,C^{\equiv}_{d}$, where $C^{\equiv}_{h}$ is the (closed) portion of $C$ between $H_{|X|+h-1}$ and $H_{|X|+h}$. We say that $C^{\equiv}_{h}$ is $h$-th fragment of $C$ and, depending on the context, may also think of $Q^{\equiv}$ as the set of fragments over all cubes. Let $S_{h}(C)$ denote $h$-th horizontal section $C\cap H_{|X|+h}$ of $C$ (where $0\leq h\leq d$); this is the convex hull of the set of vertices

[TABLE]

(Using terminology of [9], $S_{h}(C)$ is said to be a hyper-simplex. It turns into a usual simplex when $h=1$ or $d-1$.) Observe that for $h=1,\ldots,d$,

1. (6.2)

$h$-th fragment $C^{\equiv}_{h}$ of $C$ is the convex hull of the set of vertices $(X\,|\,\binom{T}{h-1})$ and $(X\,|\,\binom{T}{h}))$; it has two ”horizontal” facets, namely, $S_{h-1}(C)$ and $S_{h}(C)$, and $2d$ other facets (conditionally called “vertical” ones), namely, the portions of $F_{i}(C)$ and $G_{i}(C)$ between $H_{|X|+h-1}$ and $H_{|X|+h}$ for $i=1,\ldots,d$, denoted as $F_{h,i}(C)$ and $G_{h,i}(C)$, respectively.

Here $F_{i}(C)$ and $G_{i}(C)$ are the facets of $C=(X|T)$ defined in ((2.4)), letting $T=(p(1)<p(2)<\cdots<p(d))$. We call $S_{h-1}(C)$ and $S_{h}(C)$ the lower and upper facets of the fragment $C^{\equiv}_{h}$, respectively. Note that $S_{0}(C)$ and $S_{d}(C)$ degenerate to the single points $X$ and $XT$, respectively. The vertical facets $F_{d,i}(C)$ and $G_{1,i}(C)$ (for all $i$) degenerate as well.

The horizontal facets are “not fully seen” under the projection $\pi$. To visualize all facets of fragments of $Q^{\equiv}$, it is convenient to look at them as though “from the front and slightly from below”, i.e., by use of the projection $\pi^{\epsilon}:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d-1}$ defined by

[TABLE]

for a sufficiently small $\epsilon>0$. (Compare $\pi^{\epsilon}$ with $\pi$.) Figure 1 illustrates the case $d=3$; here the fragments of a cube $C=(X\,|\,T)$ with $T=(i<j<k)$ are drawn.

Using this projection, we obtain slightly slanted front and rear sides of objects in $Q^{\equiv}$. More precisely, for a closed set $U$ of points in $Z=Z(n,d)$, let $U^{\epsilon,\rm fr}$ ($U^{\epsilon,\rm rear}$) be the subset of $U$ “seen” in the direction $e_{d}+\epsilon e_{1}$ (resp. $-e_{d}-\epsilon e_{1}$), where $e_{i}$ is $i$-th coordinate vector, i.e., formed by the points $x\in(\pi^{\epsilon})^{-1}(x^{\prime})\cap U$ with $x_{d}$ minimum (resp. maximum) for all $x^{\prime}\in\pi^{\epsilon}(U)$. We call it the $\epsilon$-front (resp. $\epsilon$-rear) side of $U$.

Obviously, $Z^{\epsilon,\rm fr}=Z^{\,\rm fr}$ and $Z^{\epsilon,\rm rear}=Z^{\,\rm rear}$. Also for a cube $C=(X|T)$ in $Z$, $C^{\epsilon,\rm fr}=C^{\,\rm fr}$ and $C^{\epsilon,\rm rear}=C^{\,\rm rear}$. As to fragments of $C$, their $\epsilon$-front and $\epsilon$-rear sides are viewed as follows:

1. (6.4)

for $h=1,\ldots,d$,  $C^{\epsilon,\rm fr}_{h}$ is the union of $C^{\,\rm fr}_{h}$ and the lower facet $S_{h-1}(C)$ (degenerating to the point $X$ when $h=1$); in its turn, $C^{\epsilon,\rm rear}_{h}$ is formed by the union of $C^{\,\rm rear}_{h}$ and the upper facet $S_{h}(C)$ (degenerating to the point $XT$ when $h=d$).

So $C^{\epsilon,\rm fr}_{h}\cup C^{\epsilon,\rm rear}_{h}$ is just the boundary of $C^{\equiv}_{h}$.

6.2 W-membranes

Membranes of this sort represent certain $(d-1)$-dimensional subcomplexes of $Q^{\equiv}$. To introduce them, we consider small deformations of cyclic zonotopes in ${\mathbb{R}}^{d-1}$ using the projection $\pi^{\epsilon}$. More precisely, given a cyclic configuration $\Xi=(\xi_{1},\ldots,\xi_{n})$ as in ((2.1)), define

[TABLE]

Then $\Psi=(\psi_{1},\ldots,\psi_{n})$ obeys ((2.1)) (with $d-1$ instead of $d$), and when $\epsilon$ is small enough, $\Psi^{\epsilon}=(\psi^{\epsilon}_{1},\ldots,\psi^{\epsilon}_{n})$ obeys the condition ((2.1))(b), though slightly violates ((2.1))(a); yet we keep the term “cyclic configuration” for $\Psi^{\epsilon}$ as well. Consider the zonotope in ${\mathbb{R}}^{d-1}$ generated by $\Psi^{\epsilon}$, denoted as $Z^{\epsilon}(n,d-1)$ (when it is not confusing).

Definition. A w-membrane of a cubillage $Q$ on $Z(n,d)$ is a subcomplex $M$ of the fragmentation $Q^{\equiv}$ such that $M$ (regarded as a subset of ${\mathbb{R}}^{d}$) is bijectively projected by $\pi^{\epsilon}$ to $Z^{\epsilon}(n,d-1)$.

A w-membrane $M$ has facets (of dimension $d-1$) of two sorts, called H-tiles and V-tiles. Each H-tile is a horizontal facet of some fragment (viz. the section $S_{h}(C)$ of a cube $C$ in $Q$ at height $h\in[d-1]$). And V-tiles are vertical facets of some fragments $C^{\equiv}_{h}$ (see ((6.2))).

6.3 Acyclicity and the lattice structure of w-membranes

Let ${\bf C}(n,d)$ denote the set of all cubes in $Z(n,d)$ (occurring in all cubillages there). For $C,C^{\prime}\in{\bf C}(n,d)$, we say that $C$ immediately precedes $C^{\prime}$ if $C^{\,\rm rear}$ and $(C^{\prime})^{\rm fr}$ have a common facet. As a far generalization of the known acyclicity property for cubes in a cubillage, one can show the following

Proposition 6.1

The directed graph $\Gamma_{n,d}$ whose vertices are the cubes in ${\bf C}(n,d)$ and whose edges are the pairs $(C,C^{\prime})$ of cubes such that $C$ immediately precedes $C^{\prime}$ is acyclic.

(As a consequence, the transitive closure of this “immediately preceding” relation forms a partial order on ${\bf C}(n,d)$.) This proposition enables us to construct a partial order on the set of fragments for a cubillage $Q$, which in turn is used to show that the set of w-membranes in $Q^{\equiv}$ forms a distributive lattice.

More precisely, given a cubillage $Q$ on $Z(n,d)$, consider fragments $\Delta=C^{\equiv}_{i}$ and $\Delta^{\prime}=(C^{\prime})^{\equiv}_{j}$ of $Q^{\equiv}$. Let us say that $\Delta$ immediately precedes $\Delta^{\prime}$ if the $\epsilon$-rear side of $\Delta$ and the $\epsilon$-front side of $\Delta^{\prime}$ share a facet. In other words, either $C\neq C^{\prime}$ and $\Delta^{\rm rear}\cap(\Delta^{\prime})^{\rm fr}$ is a V-tile, or $C=C^{\prime}$ and $j=i+1$. The following is important for us.

Proposition 6.2

The directed graph $\Gamma_{Q^{\equiv}}$ whose vertices are the fragments in $Q^{\equiv}$ and whose edges are the pairs $(\Delta,\Delta^{\prime})$ of fragments such that $\Delta$ immediately precedes $\Delta^{\prime}$ is acyclic.

Proofs of Propositions 6.1 and 6.2 will be given in Appendix A.

From Proposition 6.2 it follows that the transitive closure of the above relation on the fragments of $Q^{\equiv}$ forms a partial order; denote it as $(Q^{\equiv},\prec)$.

Next we associate with a w-membrane $M$ of $Q$ the set $Q^{\equiv}(M)$ of fragments in $Q^{\equiv}$ lying in the region of $Z(n,d)$ between $Z^{\,\rm fr}$ and $M$. The constructions of $\pi^{\epsilon}$ and $M$ lead to the following property: for fragments $\Delta,\Delta^{\prime}$ of $Q^{\equiv}$, if $\Delta$ immediately precedes $\Delta^{\prime}$ and if $\Delta^{\prime}\in Q^{\equiv}(M)$, then $\Delta\in Q^{\equiv}(M)$ as well. This implies a similar property for fragments $\Delta,\Delta^{\prime}$ with $\Delta\prec\Delta^{\prime}$. So $Q^{\equiv}(M)$ is an ideal of $(Q^{\equiv},\prec)$. One can check that a converse property is also true: any ideal of $(Q^{\equiv},\prec)$ is expressed as $Q^{\equiv}(M)$ for some w-membrane $M$ of $Q$. Therefore,

1. (6.5)

the set ${{\cal M}^{\rm w}}(Q)$ of w-membranes of a cubillage $Q$ on $Z=Z(n,d)$ is a distributive lattice in which for $M,M^{\prime}\in{{\cal M}^{\rm w}}(Q)$, the w-membranes $M\wedge M^{\prime}$ and $M\vee M^{\prime}$ satisfy $Q^{\equiv}(M\wedge M^{\prime})=Q^{\equiv}(M)\cap Q^{\equiv}(M^{\prime})$ and $Q^{\equiv}(M\vee M^{\prime})=Q^{\equiv}(M)\cup Q^{\equiv}(M^{\prime})$; the minimal and maximal elements of this lattice are $Z^{\,\rm fr}$ and $Z^{\,\rm rear}$, respectively.

Suppose that $M\in{{\cal M}^{\rm w}}(Q)$ is different from $Z^{\,\rm fr}$; then $Q^{\equiv}(M)\neq\emptyset$. Take a maximal (relative to the order $\prec$ in $Q^{\equiv}$) fragment $\Delta$ in $Q^{\equiv}(M)$. Then $\Delta^{\epsilon,\rm rear}$ is entirely contained in $M$. Indeed, if a facet $F\in\Delta^{\epsilon,\rm rear}$ lies in $Z^{\,\rm rear}$, then $F$ is automatically in $M$. And if $F$ is not in $Z^{\,\rm rear}$, then $F$ is shared by $\Delta^{\epsilon,\rm rear}$ and $(\Delta^{\prime})^{\epsilon,\rm fr}$ for another fragment $\Delta^{\prime}$. Hence $\Delta$ immediately precedes $\Delta^{\prime}$, implying that $\Delta^{\prime}$ lies in the region between $M$ and $Z^{\,\rm rear}$. Then $F$ is in $M$, as required.

For $\Delta$ as above, the set $Q^{\equiv}(M)-\{\Delta\}$ is again an ideal of $(Q^{\equiv},\prec)$, and therefore it is expressed as $Q^{\equiv}(M^{\prime})$ for a w-membrane $M^{\prime}$. Moreover, $M^{\prime}$ is obtained from $M$ by replacing the disk $\Delta^{\epsilon,\rm rear}$ by $\Delta^{\epsilon,\rm fr}$. We call the transformation $M\mapsto M^{\prime}$ the lowering flip in $M$ using $\Delta$, and call the reverse transformation $M^{\prime}\mapsto M$ the raising flip in $M^{\prime}$ using $\Delta$. As a result, we obtain the following nice property.

Corollary 6.3

Let $M$ be a w-membrane of a cubillage $Q$. Then there exists a sequence of w-membranes $M_{0},M_{1},\ldots,M_{k}\in{{\cal M}^{\rm w}}(Q)$ such that $M_{0}=Z^{\,\rm fr}$, $M_{k}=M$, and for $i=1,\ldots,k$, $M_{i}$ is obtained from $M_{i-1}$ by the raising flip using some fragment in $Q^{\equiv}$.

6.4 Weakly $r$-separated collections via w-membranes

Now we throughout assume that $r$ is odd and $d=r+2$. Consider a cubillage $Q$ on $Z=Z(n,d)$. Based on Theorem 1.2 and Corollary 6.3, we establish the main result of Sect. 6.

Theorem 6.4

For any w-membrane $M$ of a cubillage $Q$ on $Z(n,d)$, the set $V(M)$ of vertices of $M$ (regarded as subsets of $[n]$) constitutes a maximal by size weakly $r$-separated collection in $2^{[n]}$ (where, as before, $r$ is odd and $d=r+2$). In particular, all w-membranes in $Q$ have the same number of vertices, namely, $w_{n,d-2}$ ($=s_{n,d-2}$).

Proof  Let $M\in{{\cal M}^{\rm w}}(Q)$ and consider a sequence $Z^{\,\rm fr}=M_{0},M_{1},\ldots,M_{k}=M$ as in Corollary 6.3. Let $\Delta_{1},\ldots,\Delta_{k}$ be the fragments of $Q$ such that $M_{i}$ is obtained from $M_{i-1}$ by the raising flip using $\Delta_{i}$. The collection $V(Z^{\,\rm fr})$ is weakly $r$-separated (as it is strongly $r$-separated, cf. ((3.1))), and our aim is to show that if $V(M_{i-1})$ is weakly $r$-separated, then so is $V(M_{i})$.

To show this, consider w-membranes $M,M^{\prime}$ of $Q$ such that $M^{\prime}$ is obtained from $M$ by the raising flip using a fragment $\Delta\in Q^{\equiv}$. Let $\Delta=C^{\equiv}_{h}$ for a cube $C=(X\,|\,T=(p(1)<\ldots<p(d)))$ and $h\in[d]$. By explanations in Sect. 3, $C^{\,\rm fr}$ and $C^{\,\rm rear}$ differ by exactly two vertices; namely, $V(C^{\,\rm fr})=V(C^{\,\rm rim})\cup\{t_{C}\}$ and $V(C^{\,\rm rear})=V(C^{\,\rm rim})\cup\{h_{C}\}$, where $t_{C}=Xp(2)p(4)\ldots p({d-1})$ and $h_{C}=Xp(1)p(3)\ldots p(d)$ (cf. ((3.3))). Define $R$ to be the set of vertices of $C^{\,\rm rim}$ occurring in $\Delta$, and let $r^{\prime}:=(d-1)/2$. We consider three cases.

Case 1:  $h\leq r^{\prime}$. Since the vertices of $\Delta$ are formed by the sections $S_{h-1}(C)$ and $S_{h}(C)$,

[TABLE]

(cf. (6.1)). Also $V(\Delta^{\,\rm fr})\subseteq V(\Delta^{\epsilon,\rm fr})$ and $V(\Delta^{\,\rm rear})\subseteq V(\Delta^{\epsilon,\rm rear})$. If $h<r^{\prime}$, then all vertices of $\Delta$ belong to $C^{\,\rm rim}$; this implies $V(\Delta^{\epsilon,\rm fr})=R=V(\Delta^{\epsilon,\rm rear})$. And if $h=r^{\prime}$, then the only vertex of $\Delta$ not in $R$ is $t_{C}$. Since $t_{C}\in V(C^{\,\rm fr})$, $t_{C}$ belongs to $\Delta^{\epsilon,\rm fr}$. But $t_{C}$ also lies in the upper facet $S_{r^{\prime}}(C)$ (in view of $|p(2)p(4)\ldots p(d-1)|=r^{\prime}$), and this facet is included in $\Delta^{\epsilon,\rm rear}$. Hence $t_{C}\in\Delta^{\epsilon,\rm fr}\cap\Delta^{\epsilon,\rm rear}$, implying $V(\Delta^{\epsilon,\rm fr})=V(\Delta^{\epsilon,\rm rear})$.

Case 2:  $h\geq r^{\prime}+2$. This is “symmetric” to the previous case. If $h>r^{\prime}+2$, then all vertices of $\Delta$ belong to $C^{\,\rm rim}$, implying $V(\Delta^{\epsilon,\rm fr})=R=V(\Delta^{\epsilon,\rm rear})$. And if $h=r^{\prime}+2$, then $\Delta^{\epsilon,\rm fr}$ includes the lower facet $S_{r^{\prime}+1}(C)$, which in turn contains the vertex $h_{C}$ (since $|p(1)p(3)\ldots p(d)|=r^{\prime}+1$). Also $h_{C}\in V(C^{\,\rm rear})$ implies $h_{C}\in V(\Delta^{\epsilon,\rm rear})$, and we again obtain $V(\Delta^{\epsilon,\rm fr})=V(\Delta^{\epsilon,\rm rear})$.

Thus, in both cases the raising flip $M\mapsto M^{\prime}$ using $\Delta$ does not change the vertex set of the w-membrane in question.

Case 3:  $h=r^{\prime}+1$. This case is most important. Now the lower facet $S_{h-1=r^{\prime}}(C)$ of $\Delta$ contains $t_{C}$, while the upper facet $S_{h=r^{\prime}+1}(C)$ contains $h_{C}$. Hence $t_{C}\in V(\Delta^{\epsilon,\rm fr})$ and $h_{C}\in V(\Delta^{\epsilon,\rm rear})$. On the other hand, neither $t_{C}$ belongs to $\Delta^{\epsilon,\rm rear}$ ($=\Delta^{\,\rm rear}\cup S_{r^{\prime}+1}(C)$), nor $h_{C}$ belongs to $\Delta^{\epsilon,\rm fr}$ ($=\Delta^{\,\rm fr}\cup S_{r^{\prime}}(C)$).

It follows that $V(\Delta^{\epsilon,\rm rear})=(V(\Delta^{\epsilon,\rm fr})-\{t_{C}\})\cup\{h_{C}\}$, and therefore the raising flip $M\mapsto M^{\prime}$ using $\Delta$ replaces $t_{C}$ by $h_{C}$, while preserving the other vertices of the w-membrane. Note also that the vertices of $\Delta$ different from $t_{C}$ and $h_{C}$ form just the collection of sets $XS$ such that $S$ runs over ${\cal N}(\widetilde{P},\widetilde{Q})$, the set of neighbors of $\widetilde{P}:=p(2)p(4)\ldots p(d-1)$ and $\widetilde{Q}:=p(1)p(3)\ldots p(d)$.

Now applying Theorem 1.2 to ${\cal W}:=V(M),\,X,\widetilde{P},\widetilde{Q}$ and $U:=X\widetilde{P}$, we conclude that ${\cal W}(M^{\prime})$ is weakly $r$-separated, as required.

This completes the proof of the theorem.

It should be noted that any w-membrane in a cubillage on $Z(n,3)$ can be expressed as a quasi-combined tiling in the planar zonogon $Z(n,2)$, and in this particular case, the statement of Theorem 6.4 with $r=1$ is equivalent to Theorem 3.4 in [3].

Also Theorem 6.4 together with ((6.5)) implies the following property of the set ${\bf W}^{\ast}_{n,r}$ of representable maximal by size weakly $r$-separated collections in $2^{[n]}$ (defined in the Introduction).

Corollary 6.5

${\bf W}^{\ast}_{n,r}$* is a poset with the unique minimal element $V(Z^{\,\rm fr}(n,r+2))$ and the unique maximal element $V(Z^{\,\rm rear}(n,r+2))$ in which any two neighboring elements are linked by a (raising or lowering) combinatorial flip.*

A natural question is whether any two members of the set ${\bf W}^{=}_{n,r}$ (including ${\bf W}^{\ast}_{n,r}$) can be connected by a sequence of flips. This is strengthened in the following

Conjecture 1  Let $r$ be odd and $n>r+1$. Then any maximal by size weakly $r$-separated collection ${\cal W}\subseteq 2^{[n]}$ is representable, i.e., there exists a cubillage $Q$ on $Z(n,r+2)$ and a w-membrane $M$ of (the fragmentation of) $Q$ such that $V(M)={\cal W}$.

This together with Theorem 6.4 would imply ${\bf W}^{\ast}_{n,r}={\bf W}^{=}_{n,r}$. The above assertion has been proved for $r=1$; see Theorem 3.5 in [3].

7 The non-purity phenomenon for the weak $r$-separation

Suppose that $R$ is a symmetric binary relation on elements of a set $N$ and let $G$ be the graph whose vertices are the elements of $N$ and whose edges are the pairs $\{u,v\}$ of distinct vertices subject to $uRv$. Let ${\cal C}$ be the set of cliques in $G$ (where a clique is meant to be an inclusion-wise maximal subset of vertices of which any two are connected by edge). Then ${\cal C}$ is said to be pure if all cliques of $G$ have the same size.

Recall that for an odd $r$ and $n>r$,  ${\bf W}_{n,r}$ denotes the set of all maximal by inclusion weakly $r$-separated collections in $2^{[n]}$. It was shown in [2] that ${\bf W}_{n,1}$ is pure for any $n$ (which affirmatively answers Leclerc-Zelevinsky’s conjecture on maximal weakly separated set-systems in [8]). In other words, ${\bf W}_{n,1}={\bf W}^{=}_{n,1}$ (=${\bf W}^{\ast}_{n,1}$).

In this section we show that ${\bf W}_{n,r}$ need not be pure when $n=6$ and $r=3$. In fact, we borrow a construction from [5, Sect. 3] where it is used to demonstrate the non-purity behavior for strongly $3$-separated set-systems. (Note that by a general result due to Galashin and Postnikov [7],

1. (7.1)

the set ${\bf S}_{n,r^{\prime}}$ of all inclusion-wise maximal strongly $r^{\prime}$-separated collections in $2^{[n]}$ is pure if and only if $\min\{r^{\prime},n-r^{\prime}\}\leq 2$, among all integers $r^{\prime},n$;

so $(n,r^{\prime})=(6,3)$ is the smallest case when the non-purity of ${\bf S}_{n,r^{\prime}}$ happens.)

To construct a non-pure set-system of our interest, consider the zonotope $Z=Z(6,4)$. Note that the set $V(Z)$ of vertices of (the boundary of) $Z$ consists of all intervals and all 2-intervals containing 1 or 6. (This relies on two observations: (a) any $A\subseteq[6]$ is a vertex of some cubillage in $Z$, and therefore $V(Z)\cup\{A\}$ is (strongly) 3-separated, by ((2.2)); and (b) the intervals and the 2-intervals containing 1 or 6 are exactly those subsets of $[6]$ that are $r^{\prime}$-interlaced with any subset of $[6]$, where $r^{\prime}\leq 4$.)

A direct enumeration shows that $|V(Z)|=52$. Therefore, $2^{6}-52=12$ subsets of the set $[6]$ are not in $V(Z)$; these are:

1. (7.2)

24, 245, 25, 235, 35, 135, 1356, 136, 1346, 146, 1246, 246.

(Recall that $a\cdots b$ stands for $\{a,\ldots,b\}$.) Let $A_{i}$ denote $i$-th member in this sequence (so $A_{1}=24$ and $A_{12}=246$). Form the collection

[TABLE]

It consists of $52+3=55$ sets, whereas the number $s_{6,3}=w_{6,3}$ is equal to $\binom{6}{0}+\binom{6}{1}+\binom{6}{2}+\binom{6}{3}+\binom{6}{4}=57$. Now the non-purity of ${\bf W}_{6,3}$ is implied by the following

Lemma 7.1

${\cal A}$* is a maximal weakly 3-separated collection in $2^{[6]}$.*

Proof  As mentioned above, any two sets $X\in V(Z)$ and $Y\in{\cal A}$ are 3-separated, and therefore they are weakly 3-separated. Observe that $|A_{i-1}\triangle A_{i}|=1$ for any $1\leq i\leq 12$ (where $A_{0}:=A_{12}$ and $A\triangle B$ stands for $(A-B)\cup(B-A)$). Then any $A,A^{\prime}\in\{A_{1},A_{5},A_{9}\}$ satisfy $|A\triangle A^{\prime}|\leq 4$. This implies that $A$ and $A^{\prime}$ are 3-separated. Therefore, the collection ${\cal A}$ is weakly 3-separated.

The maximality of ${\cal A}$ follows from the observation that adding to ${\cal A}$ any member of $\{A_{i}:1\leq i\leq 12,\;i\neq 1,5,9\}$ would violate the weak 3-separation. Indeed, a routine verification shows that $A_{1}$ is not weakly 3-separated from any of $A_{6},A_{7},A_{8}$, and similarly for $A_{5}$ and $\{A_{10},A_{11},A_{12}\}$, and for $A_{9}$ and $\{A_{2},A_{3},A_{4}\}$.

Remark 1.  To visualize a verification in the above proof, it is convenient to use the circular diagram in Fig. 2 where the sets from the sequence in ((7.2)) are disposed in the cyclic order. Here the sets $A_{1},A_{5},A_{9}$ are drawn in boxes and connected by lines with those sets where the weak 3-separation is violated.

In conclusion note that in light of the complete characterization for the strong separation case in ((7.1)), it is tempting to characterize all pairs $(n,r)$ (with $r$ odd) for which ${\bf W}_{n,r}$ is pure. In particular, this is so when $r=1$ (by [2]), and it is not difficult to check that ${\bf W}_{n,r}$ is pure if $n-r\leq 2$. We conjecture that the remaining cases of $(n,r)$ give the non-purity (similarly to ((7.1))), i.e., that for an odd $r$ and $n>r$, ${\bf W}_{n,r}$ is pure if and only if $\min\{r,n-r\}\leq 2$.

Appendix A Proofs of two propositions on acyclicity

Proof of Proposition 6.1  Let $C$ immediately precede $C^{\prime}$, and let the cubes $C$, $C^{\prime}$ and the facet $F:=C^{\,\rm rear}\cap(C^{\prime})^{\rm fr}$ be of the form $(X\,|\,T)$, $(X^{\prime}\,|\,T^{\prime})$ and $(\widetilde{X}\,|\,\widetilde{T})$, respectively. Then $T=\widetilde{T}\alpha$ and $T^{\prime}=\widetilde{T}\beta$ for some $\alpha,\beta\in[n]$. Four cases are possible (as illustrated in Fig. 3):

(i)  $X=X^{\prime}=\widetilde{X}$;

(ii)  $X,X^{\prime},\widetilde{X}$ are different (then $\widetilde{X}=X\alpha=X^{\prime}\beta$);

(iii)  $X\neq X^{\prime}=\widetilde{X}$ (then $\widetilde{X}=X\alpha$);

(iv)  $X^{\prime}\neq X=\widetilde{X}$ (then $\widetilde{X}=X^{\prime}\beta$).

Let us associate with a cube $C^{\prime\prime}=(X^{\prime\prime}\,|\,T^{\prime\prime})$ a label $\omega(C^{\prime\prime})\in\{0,1,2\}$ by the following rule:

• ($\ast$)

$\omega(C^{\prime\prime})=0$  if  $n\neq X^{\prime\prime},T^{\prime\prime}$;  $\omega(C^{\prime\prime})=1$  if  $n\in T^{\prime\prime}$;  $\omega(C^{\prime\prime})=2$  if  $n\in X^{\prime\prime}$.

The following observation is the key.

Claim  For $C,C^{\prime}$ as above, $\omega(C)\leq\omega(C^{\prime})$.

Proof of the Claim   We may assume that $\omega(C)\neq\omega(C^{\prime})$. Then $n\in T\cup X\cup T^{\prime}\cup X^{\prime}$ but $n$ belongs to neither $\widetilde{T}$ nor $X\cap X^{\prime}$. This implies that either $\alpha=n$ or $\beta=n$ (in view of $\widetilde{T}=T-\alpha=T^{\prime}-\beta$). Note that, as explained in Sect. 3 (when proving ((3.2))),

1. (A.1)

for a cube $C$, a facet $F_{i}(C)$ ($G_{i}(C)$) is in $C^{\,\rm fr}$ if and only if $d-i$ is even (resp. odd).

Using this for $C$ and $F$ as above and considering the inclusion $F\subset C^{\,\rm rear}$, one can conclude that if $\alpha=n$, then the root $\widetilde{X}$ of $F$ and the root $X$ of $C$ are different (taking into account that $n$ is the maximal element in $T$). In its turn, $F\subset(C^{\prime})^{\rm fr}$ implies that if $\beta=n$, then $\widetilde{X}=X^{\prime}$. This leads to the following:

1. (A.2)

$\alpha=n$ is possible only in cases (ii) and (iii), whereas $\beta=n$ is possible only in cases (i) and (iii).

In particular, case (iv) is impossible at all (when $\omega(C)\neq\omega(C^{\prime})$). As to the other three cases, we obtain from ((A.2)) that

• (a)

in case (i), $\omega(C)=0$ and $\omega(C^{\prime})=1$ (since $n=\beta\in T^{\prime}$);

• (b)

in case (ii), $\omega(C)=1$ (since $n=\alpha\in T$) and $\omega(C^{\prime})=2$ (since $\widetilde{X}=X\alpha=X^{\prime}\beta$ implies $\alpha\in X^{\prime}$);

• (c)

in case (iii), if $\alpha=n$ then $\omega(C)=1$ and $\omega(C^{\prime})=2$ (since $X^{\prime}=X\alpha$), and if $\beta=n$ then $\omega(C)=0$ and $\omega(C^{\prime})=1$.

Thus, $\omega(C)\leq\omega(C^{\prime})$ holds in all cases, as required.

Now we finish the proof of the proposition by induction on $n$. This is trivial when $n=d$, so assume that $n>d$ and that the assertion is valid for $(n^{\prime},d^{\prime})$ with $n^{\prime}<n$.

Suppose, for a contradiction, that $\Gamma_{n,d}$ has a directed cycle ${\cal C}=(C_{0},C_{1},\ldots,C_{k}=C_{0})$ (where each $C_{i}$ immediately precedes $C_{i+1}$). Then the Claim implies that $\omega(C_{i})$ is the same number $q$ for all $i$. Consider three cases (where $C_{i}=(X_{i}\,|\,T_{i})$).

Case 1: $q=0$. Then ${\cal C}$ is a directed cycle in $\Gamma_{n-1,d}$, contrary to the inductive assumption.

Case 2: $q=2$. Define $X^{\prime}_{i}:=X_{i}-n$ and $C^{\prime}_{i}:=(X^{\prime}_{i}\,|\,T_{i})$, $i=0,\ldots,k$. Then each $C^{\prime}_{i}$ is a cube in $Z(n-1,d)$, and the sequence $C^{\prime}_{0},C^{\prime}_{1},\ldots,C^{\prime}_{k}$ forms a directed cycle in $\Gamma_{n-1,d}$; a contradiction.

Case 3: $q=1$. Define $T^{\prime}_{i}:=T_{i}-n$ and $C^{\prime}_{i}:=(X_{i}\,|\,T^{\prime}_{i})$, $i=0,\ldots,k$. Then each $C^{\prime}_{i}$ can be regarded as a cube in $Z(n-1,d-1)$ (in view of $|T^{\prime}_{i}|=d-1$). Considering ((A.1)) and using the fact that $n$ is the maximal element in $T_{i}$, one can conclude that if $(Y\,|\,U)$ is a facet in $C_{i}^{\rm fr}$, then $(Y\cap[n-1]\,|\,U\cap[n-1])$ is a facet in $(C^{\prime}_{i})^{\rm rear}$, and similarly for facets in $C_{i}^{\rm rear}$ and $(C^{\prime}_{i})^{\rm fr}$. Then the fact that $C^{\,\rm rear}_{i}\cap C^{\,\rm fr}_{i+1}$ is a facet implies that $(C^{\prime}_{i})^{\rm fr}\cap(C^{\prime}_{i+1})^{\rm rear}$ is a facet as well. This means that $C^{\prime}_{i+1}$ immediately precedes $C^{\prime}_{i}$. Therefore, the sequence $C^{\prime}_{k},C^{\prime}_{k-1},\ldots,C^{\prime}_{1},C^{\prime}_{0}$ forms a directed cycle in $\Gamma_{n-1,d-1}$; a contradiction.

This completes the proof of the proposition.

Proof of Proposition 6.2  For a fragment $\Delta=C^{\equiv}_{h}$ of a cube $C=(X\,|\,T)$, denote $|X|+h-1/2$ by $\ell(\Delta)$, called the height of $\Delta$.

Suppose that there exist fragments $\Delta_{0},\Delta_{1},\ldots,\Delta_{k}=\Delta_{0}$ forming a directed cycle in $\Gamma_{Q^{\equiv}}$. Consider two consecutive fragments $\Delta=\Delta_{i-1}$ and $\Delta^{\prime}=\Delta_{i}$. Then the sides $\Delta^{\epsilon,\rm rear}$ and $\Delta^{\epsilon,\rm fr}$ share a facet $F$, and either (a) $F$ is a vertical facet of both (in terminology of ((6.2))), or (b) $F$ is the upper facet of $\Delta$ and the lower facet of $\Delta^{\prime}$. Obviously, $\ell(\Delta^{\prime})=\ell(\Delta)$ in case (a), and $\ell(\Delta^{\prime})=\ell(\Delta)+1$ in case (b). This implies

[TABLE]

Then all fragments $\Delta_{i}$ have the same height, and therefore each pair of consecutive fragments shares a vertical facet. But this means that the sequence of cubes containing these fragments forms a cycle in the graph $\Gamma_{n,d}$, contrary to Proposition 6.1.

Appendix B A concept of weak $r$-separation when $r$ is even

Up to now, we have dealt with the weak $r$-separation when $r$ is odd. In this section we attempt to introduce and explore an analogous concept when $r$ is even.

For $A^{\prime},B^{\prime}\subseteq[n]$, we say that $A^{\prime}$ surrounds $B^{\prime}$ from the right if $\max(A^{\prime}-B^{\prime})>\max(B^{\prime}-A^{\prime})$.

Definition.  For an even integer $r>0$ and an integer $n>r$, sets $A,B\subseteq[n]$ are called weakly $r$-separated if they are $\widetilde{r}$-interlaced with $\widetilde{r}\leq r+2$, and in case $\widetilde{r}=r+2$, either (a) $A$ surrounds $B$ from the right and $|A|\leq|B|$, or (b) $B$ surrounds $A$ from the right and $|B|\leq|A|$. Accordingly, a set-system ${\cal W}\subseteq 2^{[n]}$ is called weakly $r$-separated if any two members of ${\cal W}$ are such.

(Note that this matches the definition for $r$ odd in the Introduction.)

Remark 2.  In contrast to the odd case, the size $|{\cal W}|$ of a weakly $r$-separated collection ${\cal W}\subseteq 2^{[n]}$ with $r$ even can exceed the value $s_{n,r}$ (defined in ((1.2))). The simplest example is given by $n=r+2$ and ${\cal W}=2^{[n]}$. Indeed, in this case $s_{n,r}$ amounts to $\sum(\binom{r+2}{i}\,\colon\,i=0,\ldots,r+1)=2^{r+2}-1$, which is less than $|{\cal W}|=2^{r+2}$. Observe that ${\cal W}$ has only one pair $\{A,B\}$ of $(r+2)$-interlaced sets, namely, $A=\{2,4,\ldots,r\}$ and $B=\{1,3,\ldots,r-1\}$. These $A,B$ are weakly $r$-separated since $|A|=|B|$. (By the way, one can see that this ${\cal W}$ represents the vertex set of a w-membrane in the fragmentation of the cube $C=(\emptyset\,|\,[n])$, forming the trivial cubillage on $Z(n,n)$.)

Thus, if one wished to get rid of exceeding $s_{n,r}$, one should impose additional restrictions on ${\cal W}$. The above example prompts an idea on this way.

Definition.  Let us say that a pair $\{A,B\}$ of subsets of $[n]$ is a double $r$-comb if $A,B$ are $(r+2)$-interlaced and $|A\triangle B|=r+2$, i.e., $A\triangle B$ consists of elements $a_{1}<a_{2}<\ldots<a_{r+2}$ of $[n]$, one of $A-B$ and $B-A$ is formed by $a_{1},a_{3},\ldots a_{r+1}$, and the other by $a_{2},a_{4},\ldots,a_{r+2}$.

A result involving a double $r$-comb and corresponding neighbors is presented in the theorem below. This is in the spirit of Theorem 5.1 (concerning an odd $r$), but now the situation becomes more intricate.

More precisely, let $r^{\prime}:=r/2+1$, where $r$ is even as before. Let $P=\{p_{1},\ldots,p_{r^{\prime}}\}$ and $Q=\{q_{1},\ldots,q_{r^{\prime}}\}$ consist of elements of $[n]$ such that $p_{1}<q_{1}<\ldots<p_{r^{\prime}}<q_{r^{\prime}}$. Define the sets ${\cal N}^{\uparrow}(P,Q)$ and ${\cal N}^{\downarrow}(P,Q)$ of neighbors of $P,Q$ in the same way as in (5.1) and (5.2) (where now $P,Q$ satisfy $|P|=|Q|=r^{\prime}$). Let $X\subseteq[n]-(P\cup Q)$ and let $Y\subseteq[n]$ be different from $XP$ and $XQ$. We assert the following.

Theorem B.1

Let $r,n,r^{\prime},P,Q,X,Y$ be as above.

1*. Suppose that $Y$ and $XS$ are weakly $r$-separated for any $S\in{\cal N}^{\uparrow}(P,Q)$, but $Y$ and $XP$ are not. If, in addition, $Y$ and $XP$ are $(r+2)$-interlaced, then*

1. (B.1)

$Y=XQ\cup\{a\}$* for an element $a\in[n]-XPQ$ such that $a>p_{1}$.*

2*. Suppose that $Y$ and $XS$ are weakly $r$-separated for any $S\in{\cal N}^{\downarrow}(P,Q)$, but $Y$ and $XQ$ are not. If, in addition, $Y$ and $XQ$ are $(r+2)$-interlaced, then*

1. (B.2)

$Y=XP-b$* for some $b\in X$ such that $b>p_{1}$).*

(One can check that $Y$ as in ((B.1)) (resp. ((B.2))) is indeed weakly $r$-separated from any member of ${\cal N}^{\uparrow}(P,Q)$ (resp. ${\cal N}^{\downarrow}(P,Q)$) but not from $XP$ (resp. $XQ$), and the essence of the theorem is that there is no other $Y\neq XP,XQ$ with such a property.)

Proof  Let us prove assertion 1.

Analyzing the proof of Lemma 5.2, one can realize that it remains valid for corresponding $P,Q,X,Y$ when $r$ is even as well. We further rely on this lemma, borrowing, with a due care, terminology and notation from Section 5. In particular: when sets $A,B$ are not weakly $r$-separated, the pair $\{A,B\}$ is called bad; ${\cal I}$ abbreviates ${\cal I}(Y,XP)$; the intervals in ${\cal I}$ are viewed as $\ldots<A_{i-1}<B_{i}<A_{i}<B_{i+1}\ldots$ , where $A_{i^{\prime}}$ ($B_{i^{\prime}}$) stands for a $Y$-brick (resp. $XP$-brick). Also we may assume that $Y\cap X=\emptyset$ (though $Y$ and $P$ need not be disjoint).

Since $Y$ and $XP$ are required be $(r+2)$-interlaced, ${\cal I}$ consists of $r^{\prime}$ $Y$-bricks and $r^{\prime}$ $XP$-bricks. So we may assume that ${\cal I}$ is viewed as either

• (V1)

$B_{1}<A_{1}<B_{2}<A_{2}<\ldots<B_{r^{\prime}}<A_{r^{\prime}}$, or

• (V2)

$A_{1}<B_{2}<A_{2}<\ldots<B_{r^{\prime}}<A_{r^{\prime}}<B_{r^{\prime}+1}$.

Next we consider two possible cases.

Case A: $(\ast\ast)$ from Lemma 5.2 is valid. Then $A_{i}=\{q_{i}\}$ for each $i$. Hence $Y-XP=Q$ and $|Y-XP|=r^{\prime}\leq|XP-Y|$ (since each $XP$-brick $B_{j}$ contains an element of $XP-Y$). Suppose that (V1) takes place. Then $Y$-surrounds $XP$ from the right. This contradicts the condition that $\{Y,XP\}$ is bad.

And if (V2) takes place, then $p_{1}\notin XP-Y$ (since $p_{1}<q_{1}$ and $\{q_{1}\}=A_{1}$). Hence $p_{1}\in Y$. For $S:=(P-p_{1})q_{1}$, the transformation $XP\mapsto XS$ replaces the first brick $\{q_{1}\}$ of ${\cal I}$ by $\{p_{1}\}$, forming the first $Y$-brick of ${\cal I}(Y,XS)$. Then $|{\cal I}(Y,XS)|=r^{\prime}+2$, $|XS|=|XP|$, $XS$ surrounds $Y$ from the right, and therefore the badness of $\{Y,XP\}$ implies that of $\{Y,XS\}$; a contradiction.

Case B: $(\ast)$ from Lemma 5.2 is valid. Then $Y\cap P=\emptyset$, and the fact that ${\cal I}$ has exactly $r^{\prime}$ $XP$-bricks implies that $X=\emptyset$; so we may ignore $X$ in what follows. If (V2) takes place, then $XP$ surrounds $Y$ from the right. Since each $P$-brick $B_{i}$ is a singleton, $|P|=r^{\prime}\leq|Y|$, contradicting the condition that $\{Y,P\}$ is bad.

Now let (V1) take place. Then $B_{i}=\{p_{i}\}$ for each $i$. Suppose that there is $q_{i}\in Q$ such that $q_{i}\notin Y$. Then either (a) $q_{i}$ lies in some $Y$-brick $A_{j}$, or (b) no brick of ${\cal I}$ contains $q_{i}$. In case (a), we have $i=j$ (in view of $p_{i}<q_{i}<p_{i+1}$ and $p_{i}<A_{i}<p_{i+1}$, letting $p_{r^{\prime}+1}:=n+1$). Moreover, $\min(A_{i})<q_{i}<\max(A_{i})$ (since both $\min(A_{i})$ and $\max(A_{i})$ are in $Y$). Taking $S:=Pq_{i}$, we obtain $|{\cal I}(Y,S)|=|{\cal I}|+2$ (since $P\mapsto S$ replaces $A_{i}$ by the $S$-brick $\{q_{i}\}$ and two $Y$-bricks). Hence $\{Y,S\}$ is bad; a contradiction.

In case (b), three subcases are possible: either (b1) $p_{i}<q_{i}<A_{i}$; or (b2) $A_{i}<q_{i}<p_{i+1}$, or (b3) $i=r^{\prime}$ and $A_{r^{\prime}}<q_{r^{\prime}}$. In these subcases, taking as $S$ the neighbors $(P-p_{i})q_{i}$, $(P-p_{i+1})q_{i}$, and $Pq_{r^{\prime}}$, respectively, one can see that $\{Y,S\}$ is bad.

Thus, $Q\subseteq Y$. Note that any $Y$-brick $A_{i}$ contains at most one element of $Q$ (for if $A_{i}$ would contain $q_{j-1}$ and $q_{j}$ say, then $A_{i}$ should contain $p_{j}$ as well, which is impossible). It follows that each $A_{i}$ contains exactly one element of $Q$, namely, $q_{i}$. Since $\{Y,P\}$ is bad and $Y$ surrounds $P$ from the right, there must be $|Y|>|P|=r^{\prime}$. So at least one $Y$-brick $A_{i}$ has size $\geq 2$. For such an $A_{i}$, taking $S:=Pq_{i}$, one can see that $|{\cal I}(Y,S)|=|{\cal I}|$ and that $Y$ surrounds $S$ from the right. Then $|Y|\leq|S|$ (otherwise $\{Y,S\}$ is bad). This together with $|Y|>r^{\prime}$ and $|S|=|P|+1=r^{\prime}+1$ gives $|Y|=r^{\prime}+1$. The latter means that there is exactly one brick $A_{i}$ of size $\geq 2$; moreover, $|A_{i}|=2$. Then $A_{i}=\{q_{i},a\}$, where $a$ is as required in ((B.1)), yielding assertion 1 of the theorem.

Assertion 2 of the theorem can be shown by symmetry and we leave details to the reader.

Remark 3. Some neighbors of $P,Q$ arising in connection with Theorem B.1 play an especial role. More precisely, let $Y=XQ\cup\{a\}$ be as in ((B.1)); then $p_{i}<a<p_{i+1}$ for some $i\in[r^{\prime}]$ (letting $p_{r^{\prime}+1}:=n+1$). One can check that in the upper neighbor collection $\{S\subset P\cup Q\,\colon\,S\neq P,Q,\;r^{\prime}\leq|S|\leq r^{\prime}+1\}$ (which includes ${\cal N}^{\uparrow}(P,Q)$) there is exactly one set $S$ such that $\{Y,XS\}$ is a double $r$-comb; this is $S=Pq_{i}$. (Then $XS-Y=\{p_{1},\ldots,p_{r^{\prime}}\}$ and $Y-XS=\{q_{1},\ldots,q_{i-1},a,q_{i+1},\ldots,q_{r^{\prime}}\}$.) Symmetrically, for $Y=XP-b$ as in ((B.2)), in the lower neighbor collection $\{S\subset P\cup Q\,\colon\,S\neq P,Q,\;r^{\prime}-1\leq|S|\leq r^{\prime}\}$ (which includes ${\cal N}^{\downarrow}(P,Q)$) there is exactly one $S$ such that $\{Y,XS\}$ is a double $r$-comb. Namely, if $p_{i}<b<p_{i+1}$ (letting $p_{r^{\prime}+1}:=n+1$), then $S=Q-q_{i}$. (In this case, $Y-XS=\{p_{1},\ldots,p_{r^{\prime}}\}$ and $XS-Y=\{q_{1},\ldots,q_{i-1},b,q_{i+1},\ldots,q_{r^{\prime}}\}$).

The rest of this section is devoted to a geometric construction representing a class of $r$-separated collections. This relies on Theorem B.1 and is in the spirit of the construction from Sect. 6.4 (with $r$ odd), though looks a bit more intricate. We will use terminology and notation from Sect. 6.

As before, let $r$ be even and $r^{\prime}=r/2+1$. For $d:=r+2$, consider a cubillage $Q$ on the zonotope $Z=Z(n,d)$ and its fragmentation $Q^{\equiv}$. For each cube $C=(X|T)\in Q$, we distinguish two “central” fragments $C^{\equiv}_{r^{\prime}}$ and $C^{\equiv}_{r^{\prime}+1}$. They share the middle horizontal section $S_{d/2}(C)$ ($=C\cap H_{|X|+r^{\prime}}$), which contains the specified vertices $t_{C}=XP$ and $h_{C}=XQ$, where $T=(p_{1}<q_{1}<\ldots<p_{r^{\prime}}<q_{r^{\prime}})$, $P=\{p_{1},\ldots,p_{r^{\prime}}\}$ and $Q=\{q_{1},\ldots,q_{r^{\prime}}\}$ (so $\{t_{C},h_{C}\}$ forms a double $r$-comb).

Definitions. For a cube $C\in Q$, the set $C^{\equiv}_{r^{\prime}}\cup C^{\equiv}_{r^{\prime}+1}$ is called the doubled fragment, or the center, of $C$ and denoted by $C^{\equiv}_{\rm cup}$; the remaining fragments $C^{\equiv}_{h}$ of $C$ ($h\neq r^{\prime},r^{\prime}+1$) are called ordinary ones. By the enlarged fragmentation of $Q$ we mean the complex generated by the centers and ordinary fragments of all cubes of $Q$, denoted as ${Q^{\equiv}_{\rm en}}$, i.e., it is obtained from $Q^{\equiv}$ by merging the pieces $C^{\equiv}_{r^{\prime}}$ and $C^{\equiv}_{r^{\prime}+1}$ into $C^{\equiv}_{\rm cup}$ for each $C\in Q$. Depending on the context, we may also think of ${Q^{\equiv}_{\rm en}}$ as the collection of doubled and ordinary fragments over all the cubes of $Q$.

This gives rise to an important subclass of w-membranes. More precisely, when a w-membrane $M$ of $Q$ is a subcomplex (of dimension $d-1$) of ${Q^{\equiv}_{\rm en}}$, we say that $M$ is an e-membrane. It is not difficult to show that a w-membrane $M$ of this sort is characterized by the property that no facet of $M$ is the middle section of a cube of $Q$, or, equivalently, that for each cube $C\in Q$, $M$ meets at most one vertex among $t_{C},h_{C}$.

Like c- and w-membranes, the set of e-membranes of $Q$ forms a distribute lattice. This is based on the following

Proposition B.2

*The directed graph $\Gamma_{{Q^{\equiv}_{\rm en}}}$ whose vertices are the fragments in ${Q^{\equiv}_{\rm en}}$ and whose edges are the pairs $(\Delta,\Delta^{\prime})$ of fragments such that $\Delta$ immediately precedes $\Delta^{\prime}$ (in the sense that $\Delta^{\epsilon,\rm rear}$ and $(\Delta^{\prime})^{\,{\epsilon,\rm fr}}$ share a facet) is acyclic.   *

Proof  This is similar to the proof of Proposition 6.2 and is briefly as follows. Suppose that fragments $\Delta_{0},\Delta_{1},\ldots,\Delta_{k}=\Delta_{0}$ of ${Q^{\equiv}_{\rm en}}$ form a directed cycle in $\Gamma_{{Q^{\equiv}_{\rm en}}}$. For each $i$, let $C_{i}$ be the cube of $Q$ containing $\Delta_{i}$. If $C_{i}=C_{i+1}$, then the height of $\Delta_{i+1}$ is greater than that of $\Delta_{i}$. Therefore, a maximal subsequence $S$ of different cubes among $C_{0},C_{1},\ldots,C_{k-1}$ consists of more than one element. Moreover, consecutive cubes in $S$ share a (vertical) facet, whence $S$ determines a directed cycle in $\Gamma_{n,d}$, contradicting Proposition 6.1.

Thus, the transitive closure of the above relation on the fragments of ${Q^{\equiv}_{\rm en}}$ forms a partial order, denoted as $\prec_{\rm en}$. As a consequence (cf. ((6.5)) and Corollary 6.3):

1. (B.3)

For an e-membrane $M$ of $Q$, let ${Q^{\equiv}_{\rm en}}(M)$ denote the collection of fragments of ${Q^{\equiv}_{\rm en}}$ lying between $Z^{\,\rm fr}$ and $M$. Then the set ${{\cal M}^{\rm e}}(Q)$ of e-membranes of a cubillage $Q$ on $Z(n,d)$ is a distributive lattice, with the minimal element $Z^{\,\rm fr}$ and the maximal element $Z^{\,\rm rear}$, in which for $M,M^{\prime}\in{{\cal M}^{\rm e}}(Q)$, the e-membranes $M\wedge M^{\prime}$ and $M\vee M^{\prime}$ satisfy ${Q^{\equiv}_{\rm en}}(M\wedge M^{\prime})={Q^{\equiv}_{\rm en}}(M)\cap{Q^{\equiv}_{\rm en}}(M^{\prime})$ and ${Q^{\equiv}_{\rm en}}(M\vee M^{\prime})={Q^{\equiv}_{\rm en}}(M)\cup{Q^{\equiv}_{\rm en}}(M^{\prime})$.

1. (B.4)

Let $M$ be an e-membrane of $Q$. Then there exists a sequence of e-membranes $M_{0},M_{1},\ldots,M_{k}\in{{\cal M}^{\rm e}}(Q)$ such that $M_{0}=Z^{\,\rm fr}$, $M_{k}=M$, and for $i=1,\ldots,k$, $M_{i-1}$ is obtained from $M_{i}$ by the lowering flip using some maximal (w.r.t. $\prec_{\rm en}$) fragment $\Delta$ in ${Q^{\equiv}_{\rm en}}(M_{i})$ (in the sense that $\Delta^{\epsilon,\rm rear}\subset M_{i}$, and $M_{i-1}$ is obtained from $M_{i}$ by replacing the disk $\Delta^{\epsilon,\rm rear}$ by $\Delta^{\epsilon,\rm fr}$).

Based on the above properties, we obtain a geometric result which can be viewed, to some extent, as a counterpart of Theorem 6.4 (concerning the odd case).

Theorem B.3

Let $r$ be even and $d:=r+2$. Suppose that a cubillage $Q$ on $Z=Z(n,d)$ possesses the property that

• (P)

no e-membrane of $Q$ has a pair of vertices forming a double $r$-comb.

Then for any e-membrane $M$ of $Q$,

• (i)

the set $V(M)$ of vertices of $M$ (regarded as subsets of $[n]$) is weakly $r$-separated;

• (ii)

$|V(M)|=s_{n,r}$.

Proof  We argue as in the proof of Theorem 6.4. For $M\in{{\cal M}^{\rm e}}(Q)$, consider a sequence $Z^{\,\rm fr}=M_{0},M_{1},\ldots,M_{k}=M$ of e-membranes of $Q$ as in ((B.4)). Since $Z^{\,\rm fr}$ satisfies (i),(ii), it suffices to prove the following assertion.

1. (B.5)

For $M,M^{\prime}\in{{\cal M}^{\rm e}}(Q)$, let $M^{\prime}$ be obtained from $M$ by the raising flip using a (double or ordinary) fragment $\Delta$ of ${Q^{\equiv}_{\rm en}}$, and suppose that $M$ satisfies (i),(ii). Then $M^{\prime}$ satisfies (i),(ii) as well.

To show this, assume that $\Delta$ belongs to a cube $C=(X|T)\in Q$. When $\Delta$ is ordinary, i.e., $\Delta=C^{\equiv}_{h}$ with $h\leq r^{\prime}-1$ or $h\geq r^{\prime}+2$ (where $r^{\prime}=r/2+1$), then $V(M^{\prime})=V(M)$, and we are done (cf. explanations in Cases 1 and 2 of the proof of Theorem 6.4).

So let $\Delta$ be the center $C^{\equiv}_{\rm cup}$ of $C$. The raising flip using $\Delta$ replaces in $M$ the side $\Delta^{\epsilon,\rm fr}$ by $\Delta^{\epsilon,\rm rear}$. One can see that the vertex $t_{C}$ of $C$ is in $\Delta^{\epsilon,\rm fr}$ by not in $\Delta^{\epsilon,\rm rear}$, while $h_{C}$ is in $\Delta^{\epsilon,\rm rear}$ by not in $\Delta^{\epsilon,\rm fr}$, and that the other vertices of $\Delta$ and $\Delta^{\prime}$ coincide. Therefore, the flip replaces $t_{C}=XP$ by $h_{C}=XQ$, yielding $V(M^{\prime})=(V(M)-\{t_{C}\})\cup\{h_{C}\}$, where $T=(p_{1}<q_{1}<\ldots<p_{r^{\prime}}<q_{r^{\prime}})$, $P=\{p_{1},\ldots,p_{r^{\prime}}\}$ and $Q=\{q_{1},\ldots,q_{r^{\prime}}\}$.

We have $|V(M^{\prime})|=|V(M)|$; so $M^{\prime}$ satisfies (ii). Suppose, for a contradiction, that (i) is false for $M^{\prime}$, i.e., there are two vertices of $M^{\prime}$ that are not weakly $r$-separated from each other. Then one of them is $XQ$, and the other, $Y$ say, belongs to $M$ and differs from $XP$. By (i) for $M$, the vertex $Y$ is weakly $r$-separated from $XS$ for each neighbor $S\in{\cal N}^{\downarrow}(P,Q)$. (Note that $XS$ lies in $\Delta^{\epsilon,\rm fr}$ even if $|S|=r^{\prime}$.) So we can apply assertion 2 of Theorem B.1 and conclude that $Y$ is viewed as in ((B.2)). But then, as mentioned in Remark 3, there is $S\in{\cal N}^{\downarrow}(P,Q)$ such that $\{Y,XS\}$ is a double $r$-comb (namely, $Y=XP-b$ and $S=Q-q_{i}$, where $p_{i}<b<p_{i+1}$). Hence $M$ contains a double $r$-comb, contrary to condition (P).

Thus, ((B.5)) is valid, and the theorem follows.

Remark 4.  By the construction of an e-membrane $M$ of a cubillage $Q$, $M$ has no double $r$-comb of the form $\{t_{C},h_{C}\}$ for a cube $C$ of $Q$. However, a priori it is not clear whether $M$ is free of double $r$-combs at all. We conjecture that this is so for any e-membrane, i.e., property (P) holds for any cubillage $Q$. Its validity would give a strengthening of Theorem B.3. We state it as follows:

Conjecture 2   For $r$ even, the vertex set $V(M)$ of any e-membrane $M$ of an arbitrary cubillage $Q$ on $Z(n,r+2)$ gives a weakly $r$-separated collection.

(Note that such a $V(M)$ is automatically of size $s_{n,r}$, by explanations above.) It is tempting to conjecture a sharper property (which is just a direct analog of Theorem 6.4), claiming that $V(M)$ is weakly $r$-separated for any w-membrane of a cubillage $Q$ on $Z(n,r+2)$ (where $|V(M)|$ may exceed $s_{n,r}$), but we do not go so far at the moment.

We finish with an analogue of Conjecture 1:

Conjecture 3   For $r$ even, the maximal size of a weakly $r$-separated collection $W\subseteq 2^{[n]}$ without double $r$-combs is equal to $s_{n,r}$ and such a $W$ with $|W|=s_{n,r}$ is representable, in the sense that there exists a cubillage $Q$ on $Z(n,r+2)$ and an e-membrane $M$ of $Q$ such that $V(M)=W$.