Approximate Steiner $(r-1,r,n)$-systems without $3$ blocks on $r+2$   points

Alexander Sidorenko

arXiv:1907.08084·math.CO·December 17, 2019

Approximate Steiner $(r-1,r,n)$-systems without $3$ blocks on $r+2$ points

Alexander Sidorenko

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TL;DR

This paper investigates the maximum size of certain hypergraphs avoiding specific small configurations, providing asymptotic bounds for these extremal problems in combinatorics.

Contribution

It establishes asymptotic bounds for the maximum number of edges in hypergraphs avoiding particular small subgraphs, advancing understanding of approximate Steiner systems.

Findings

01

Asymptotic maximum edges are approximately (1/r) * binomial(n, r-1).

02

Provides bounds for hypergraphs avoiding small configurations.

03

Advances extremal combinatorics in hypergraph theory.

Abstract

For a family $F$ of $r$ -graphs, let $ex (n, F)$ denote the maximum number of edges in an $F$ -free $r$ -graph on $n$ vertices. Let $F_{r} (v, e)$ denote the family of all $r$ -graphs with $e$ edges and at most $v$ vertices. We prove that $ex (n, F_{r} (r + 1, 2) \cup F_{r} (r + 2, 3)) = (\frac{1}{r} - o (1)) (r - 1 n)$ .

Equations24

(c_{r, v, e} + o (1)) n^{k} \leq f_{r} (n, v, e) \leq (C_{r, v, e} + o (1)) n^{k} as n \to \infty .

(c_{r, v, e} + o (1)) n^{k} \leq f_{r} (n, v, e) \leq (C_{r, v, e} + o (1)) n^{k} as n \to \infty .

f_{r} (n, v, 2) = ((2 r - v r)^{- 1} + o (1)) (2 r - v n) as n \to \infty .

f_{r} (n, v, 2) = ((2 r - v r)^{- 1} + o (1)) (2 r - v n) as n \to \infty .

f_{r} (n, 3 r - 4, 3) = (\frac{1}{r ^{2} - r - 1} + o (1)) n^{2} as n \to \infty .

f_{r} (n, 3 r - 4, 3) = (\frac{1}{r ^{2} - r - 1} + o (1)) n^{2} as n \to \infty .

f_{r} (n, 3 r - 2 k, 3) \leq (\frac{1}{( k r ) - \frac{1}{2}} + o (1)) (k n) as n \to \infty,

f_{r} (n, 3 r - 2 k, 3) \leq (\frac{1}{( k r ) - \frac{1}{2}} + o (1)) (k n) as n \to \infty,

f_{r} (n, r + 2, 3) \leq (\frac{1}{r - \frac{1}{2}} + o (1)) (r - 1 n) as n \to \infty .

f_{r} (n, r + 2, 3) \leq (\frac{1}{r - \frac{1}{2}} + o (1)) (r - 1 n) as n \to \infty .

f_{r} (n, r + 2, 3) \geq (\frac{1}{r} (3 \frac{1}{2} ( r + 2 ) ( r + 1 ))^{- 1/2} + o (1)) (r - 1 n) as n \to \infty .

f_{r} (n, r + 2, 3) \geq (\frac{1}{r} (3 \frac{1}{2} ( r + 2 ) ( r + 1 ))^{- 1/2} + o (1)) (r - 1 n) as n \to \infty .

f_{r} (n, r + 2, 3) \geq (\frac{1}{r} - o (1)) (r - 1 n) as n \to \infty .

f_{r} (n, r + 2, 3) \geq (\frac{1}{r} - o (1)) (r - 1 n) as n \to \infty .

\mathrm{ex}\Big{(}n,\>{\mathcal{F}}_{3}(4,2)\cup{\mathcal{F}}_{3}(5,3)\cup\ldots\cup{\mathcal{F}}_{3}(m+2,m)\Big{)}\;=\;\left(\frac{1}{3}+o(1)\right)\binom{n}{2}\;.

\mathrm{ex}\Big{(}n,\>{\mathcal{F}}_{3}(4,2)\cup{\mathcal{F}}_{3}(5,3)\cup\ldots\cup{\mathcal{F}}_{3}(m+2,m)\Big{)}\;=\;\left(\frac{1}{3}+o(1)\right)\binom{n}{2}\;.

\mathrm{ex}\Big{(}n,{\mathcal{F}}_{r}(r+1,2)\cup{\mathcal{F}}_{r}(r+2,3)\cup\ldots\cup{\mathcal{F}}_{r}(r+m-1,m)\Big{)}=\left(\frac{1}{r}+o(1)\right)\binom{n}{r-1}.

\mathrm{ex}\Big{(}n,{\mathcal{F}}_{r}(r+1,2)\cup{\mathcal{F}}_{r}(r+2,3)\cup\ldots\cup{\mathcal{F}}_{r}(r+m-1,m)\Big{)}=\left(\frac{1}{r}+o(1)\right)\binom{n}{r-1}.

1 + 1 + 1 \equiv i \in A \sum i + i \in B \sum i + i \in C \sum i \equiv 2 i \in D \cup E \cup F \sum i (mod n),

1 + 1 + 1 \equiv i \in A \sum i + i \in B \sum i + i \in C \sum i \equiv 2 i \in D \cup E \cup F \sum i (mod n),

0 = z \in A \sum z + z \in B \sum z + z \in C \sum z = 2 z \in \overline{A} \cup \overline{B} \cup \overline{C} \sum z + 3 z \in D \sum z = z \in D \sum z,

0 = z \in A \sum z + z \in B \sum z + z \in C \sum z = 2 z \in \overline{A} \cup \overline{B} \cup \overline{C} \sum z + 3 z \in D \sum z = z \in D \sum z,

\mathrm{ex}\Big{(}n,\>{\mathcal{F}}_{r}(r+1,2)\cup{\mathcal{F}}_{r}(r+2,3)\Big{)}\;=\;\left(\frac{1}{r}-o(1)\right)\binom{n}{r-1}\;.

\mathrm{ex}\Big{(}n,\>{\mathcal{F}}_{r}(r+1,2)\cup{\mathcal{F}}_{r}(r+2,3)\Big{)}\;=\;\left(\frac{1}{r}-o(1)\right)\binom{n}{r-1}\;.

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Approximate Steiner $(r-1,r,n)$ -systems

without $3$ blocks on $r+2$ points

Alexander Sidorenko

[email protected]

Abstract

For a family ${\mathcal{F}}$ of $r$ -graphs, let $\mathrm{ex}(n,{\mathcal{F}})$ denote the maximum number of edges in an ${\mathcal{F}}$ -free $r$ -graph on $n$ vertices. Let ${\mathcal{F}}_{r}(v,e)$ denote the family of all $r$ -graphs with $e$ edges and at most $v$ vertices. We prove that $\mathrm{ex}(n,{\mathcal{F}}_{r}(r+1,2)\cup{\mathcal{F}}_{r}(r+2,3))=(\frac{1}{r}-o(1))\binom{n}{r-1}$ .

Let ${\mathcal{F}}$ be a family of $r$ -graphs, (that is, $r$ -uniform hypergraphs). An $r$ -graph is called ${\mathcal{F}}$ -free if it does not have a subgraph isomorphic to a member of ${\mathcal{F}}$ . Let $\mathrm{ex}(n,{\mathcal{F}})$ denote the maximum number of edges in an ${\mathcal{F}}$ -free $r$ -graph on $n$ vertices. Let ${\mathcal{F}}_{r}(v,e)$ denote the family of all $r$ -graphs with $e$ edges and at most $v$ vertices. In 1973, Brown, Erdős and Sós initiated the study of $f_{r}(n,v,e):=\mathrm{ex}(n,{\mathcal{F}}_{r}(v,e))$ . The case $r=2$ had been studied before. In particular, Erdős [4] proved $f_{2}(n,e+1,e)=\left\lfloor\frac{e-1}{e}n\right\rfloor$ . Cases when the ratio $(er-v)/(e-1)$ is not integer were studied in [1, 6, 7, 11, 12, 13, 15]. In the case when $k=(er-v)/(e-1)$ is an integer, Brown, Erdős and Sós [3] were able to prove

[TABLE]

The case $e=2$ is equivalent to the packing problem and was asymptotically solved by Rödl [10] who proved

[TABLE]

Recently, the first case with $e=3$ , $r\geq 3$ was solved by Glock [9]: $f_{3}(n,5,3)=(\frac{1}{5}+o(1))n^{2}$ . His result was generalized by Shangguan and Tamo [14] to

[TABLE]

Shangguan and Tamo proved the upper bound

[TABLE]

which in the case $e=3$ , $\>v=r+2$ becomes

[TABLE]

A lower bound, obtained from the probabilistic construction of Brown, Erdős, and Sós [3], was explicitly calculated in [14]:

[TABLE]

We will prove a much better lower bound which almost matches the upper one:

[TABLE]

Answering a question of Erdős [5], two sets of authors in [2] and [8] independently proved that for any $m\geq 2$ ,

[TABLE]

It was suggested in [8, Conjecture 7.2] that a similar statement must hold for all $r\geq 3$ , $m\geq 2$ , namely

[TABLE]

We will prove this statement for $m=3$ by constructing ${\mathcal{F}}_{r}(r+2,3)$ -free (or, in terms of [8], $3$ -sparse) partial Steiner $(r-1,r,n)$ -systems with $(\frac{1}{r}-o(1))\binom{n}{r-1}$ blocks (edges). The case $r=3$ is trivial, because any ${\mathcal{F}}_{3}(4,2)$ -free system is also ${\mathcal{F}}_{3}(5,3)$ -free. We will provide different constructions for $r=4$ (Theorem 1) and $r\geq 5$ (Theorem 4).

We define the degree of a subset of vertices in an $r$ -graph as the number of edges that contain this subset.

Theorem 1.

For any even $n$ , there exists an $({\mathcal{F}}_{4}(5,2)\cup{\mathcal{F}}_{4}(6,3))$ -free $4$ -graph with $n$ vertices and $\frac{1}{4}\binom{n}{3}-\frac{n(n-2)}{8}$ edges.

Proof.

Consider a $4$ -graph $H(n)$ whose vertices are elements of $\mathbb{Z}_{n}$ , and $4$ distinct vertices $i,j,k,l$ form an edge when $i+j+k+l\equiv 1\pmod{n}$ . Then the degree of a triple $\{i,j,k\}$ is either $1$ (when $i+j+2k,\>i+2j+k,\>2i+j+k\not\equiv 1\pmod{n})$ , or [math]. For each $i\in\mathbb{Z}_{n}$ , there are exactly $\frac{1}{2}(n-2)$ pairs $\{j,k\}$ such that $j\neq i$ , $k\neq i$ , $j\neq k$ , and $2i+j+k\equiv 1\pmod{n}$ . Hence, the number of triples of degree [math] is $\frac{n(n-2)}{2}$ , and consequently, the number of edges in $H(n)$ is $\frac{1}{4}\Huge(\binom{n}{3}-\frac{n(n-2)}{2}\Huge)$ .

The only $4$ -graph in ${\mathcal{F}}_{4}(5,2)\cup{\mathcal{F}}_{4}(6,3)$ without triples of degree $\geq 2$ is formed by $3$ edges $A,B,C$ such that $A=D\cup E$ , $B=D\cup F$ , $C=E\cup F$ , where $D,E,F$ are disjoint pairs of vertices. If they were present in $H(n)$ , we would get

[TABLE]

which is impossible (as $n$ is even). ∎

Proposition 2.

When $r>4$ and $n=2^{d}$ , there exists an $({\mathcal{F}}_{r}(r+1,2)\cup{\mathcal{F}}_{r}(r+2,3))$ -free $r$ -graph with $n$ vertices and $\frac{1}{r}\binom{n}{r-1}-O(n^{r-2})$ edges.

Proof.

Consider an $r$ -graph $H_{d}$ whose vertices are elements of $\mathbb{Z}_{2}^{d}$ , where a set $A$ of $r$ elements is an edge if the sum of them is zero, and $A$ does not contain zero-sum subsets of sizes $4$ and $r-4$ . It is easy to see that the degree of any $(r-1)$ -subset of vertices is at most $1$ . An $(r-1)$ -subset has degree [math] only when it contains a zero-sum subset of size $r-2$ , $r-4$ or $4$ . For a fixed $k$ , the number of zero-sum subsets of size $k$ is $\frac{1+o(1)}{n}\binom{n}{k}$ , so the number of $(r-1)$ -subsets that contain a zero-sum subset of size $k$ is at most $\frac{1+o(1)}{n}\binom{n}{k}\binom{n-k}{r-1-k}=O(n^{r-2})$ . Hence, the number of $(r-1)$ -subsets of degree [math] is $O(n^{r-2})$ , and the number of edges in $H_{d}$ is $\frac{1}{r}\left(\binom{n}{r-1}-O(n^{r-2})\right)$ . Obviously, $H_{d}$ is ${\mathcal{F}}_{r}(r+1,2)$ -free. We need to show that $H_{d}$ is ${\mathcal{F}}_{r}(r+2,3)$ -free. Suppose, there exist an $(r+2)$ -subset $X$ and $3$ distinct edges $A,B,C\subset X$ . Since $|A\cup B|=|A|+|B|-|A\cap B|\geq 2r-(r-2)=r+2=|X|$ , we get $A\cup B=X$ , and similarly, $A\cup C=B\cup C=X$ . Set $\overline{A}=X-A$ , $\overline{B}=X-B$ , $\overline{C}=X-C$ . Then $|\overline{A}|=|\overline{B}|=|\overline{C}|=|X|-r=2$ , and $\overline{A}$ , $\overline{B}$ , $\overline{C}$ are pairwise disjoint, because $\overline{A}\cap\overline{B}=X-(A\cup B)=\emptyset$ . Set $D=X-(\overline{A}\cup\overline{B}\cup\overline{C})$ . Then $A=D\cup\overline{B}\cup\overline{C}$ , $B=D\cup\overline{A}\cup\overline{C}$ , $C=D\cup\overline{A}\cup\overline{B}$ , $|D|=(r+2)-3\cdot 2=r-4$ , and we get

[TABLE]

so $D$ is a zero-sum subset of size $r-4$ in edge $A$ , a contradiction. ∎

Proposition 3.

When $r>4$ and $n=m2^{d}$ , there exists an $({\mathcal{F}}_{r}(r+1,2)\cup{\mathcal{F}}_{r}(r+2,3))$ -free $r$ -graph $H_{m,d}$ with $n$ vertices and $(\frac{1}{r}-O(\frac{1}{2^{d}}))\binom{n}{r-1}$ edges.

Proof.

The case $m=1$ is covered by Proposition 2. Assume $m\geq 2$ and consider an $r$ -graph $H_{m,d}$ whose vertices are elements of $\mathbb{Z}_{m}\oplus\mathbb{Z}_{2}^{d}$ , and $A=\{(x_{1},y_{1}),(x_{2},y_{2}),\ldots,(x_{r},y_{r})\}$ is an edge in $H_{m,d}$ when $x_{1}+x_{2}+\ldots+x_{r}=0$ in $\mathbb{Z}_{m}$ and $A^{\prime}=\{y_{1},y_{2},\ldots,y_{r}\}$ is an edge in $r$ -graph $H_{d}$ from Proposition 2. (We say that $A^{\prime}$ is the shadow of $A$ in $H_{d}$ .) The number of edges in $H_{m,d}$ is $m^{r-1}$ times that of $H_{d}$ . It is easy to see that $H_{m,d}$ is ${\mathcal{F}}_{r}(r+1,2)$ -free. We need to check that it is also ${\mathcal{F}}_{r}(r+2,3)$ -free. Suppose, there exist three distinct edges $A,B,C$ such that $|A\cup B\cup C|\leq r+2$ . As $H_{d}$ is ${\mathcal{F}}_{r}(r+2,3)$ -free, at least two of these three edges must have the same shadow in $H_{d}$ . Let $A=\{(x_{1}^{\prime},y_{1}),(x_{2}^{\prime},y_{2}),\ldots,(x_{r}^{\prime},y_{r})\}$ and $B=\{(x_{1}^{\prime\prime},y_{1}),(x_{2}^{\prime\prime},y_{2}),\ldots,(x_{r}^{\prime\prime},y_{r})\}$ . Since $x_{1}^{\prime}+x_{2}^{\prime}+\ldots+x_{r}^{\prime}=0$ and $x_{1}^{\prime\prime}+x_{2}^{\prime\prime}+\ldots+x_{r}^{\prime\prime}=0$ , vectors $(x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{r}^{\prime})$ and $(x_{1}^{\prime\prime},x_{2}^{\prime\prime},\ldots,x_{r}^{\prime\prime})$ must differ in at least two places. Since $|A\cup B|\leq r+2$ , the same vectors must differ in at most two places. Therefore, they differ in exactly two places, and $|A\cup B|=r+2$ . Then $C$ must have the same shadow as $A$ and $B$ : $C=\{(x_{1}^{\prime\prime\prime},y_{1}),(x_{2}^{\prime\prime\prime},y_{2}),\ldots,(x_{r}^{\prime\prime\prime},y_{r})\}$ . Similarly, $(x_{1}^{\prime\prime\prime},x_{2}^{\prime\prime\prime},\ldots,x_{r}^{\prime\prime\prime})$ must differ in exactly two places from $(x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{r}^{\prime})$ and from $(x_{1}^{\prime\prime},x_{2}^{\prime\prime},\ldots,x_{r}^{\prime\prime})$ . As $|A\cup B\cup C|=r+2$ , it must be the same places where $A$ and $B$ differ, but this is impossible. ∎

Theorem 4.

When $r>4$ , there exists an $({\mathcal{F}}_{r}(r+1,2)\cup{\mathcal{F}}_{r}(r+2,3))$ -free $r$ -graph with $\leq n$ vertices and $(\frac{1}{r}-O(\frac{1}{\sqrt{n}}))\binom{n}{r-1}$ edges as $n\to\infty$ .

Proof.

For a given $n$ , select $d=\lfloor\log_{2}\sqrt{n}\rfloor$ and $m=\lfloor n/2^{d}\rfloor$ . Then $0\leq n-m2^{d}<2^{d}\leq\sqrt{n}$ . By Proposition 3, there exists an $({\mathcal{F}}_{r}(r+1,2)\cup{\mathcal{F}}_{r}(r+2,3))$ -free $r$ -graph with $m2^{d}$ vertices and $(\frac{1}{r}-O(\frac{1}{2^{d}}))\binom{m2^{d}}{r-1}$ edges. Notice that $\frac{1}{2^{d}}=O(\frac{1}{\sqrt{n}})$ and $\binom{m2^{d}}{r-1}=(1-O(\frac{1}{\sqrt{n}}))\binom{n}{r-1}$ as $n\to\infty$ . ∎

Any ${\mathcal{F}}_{r}(r+1,2)$ -free $r$ -graph has at most $\frac{1}{r}\binom{n}{r-1}$ edges (because every $(r-1)$ -subset of vertices has degree at most $1$ ). Hence, Theorems 1 and 4 imply

Corollary 5.

When $r\geq 3$ and $n\to\infty$ ,

[TABLE]

Simplicity of our constructions of approximate Steiner $(r-1,r,n)$ -systems (comparing with Rödl’s probabilistic proof of 1) can be explained by the fact that their main parameters, $r-1$ and $r$ , differ just by $1$ .

Acknowledgments. The author would like to thank two anonymous referees for their valuable comments and suggestions.

Bibliography15

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] N. Alon and A. Shapira. On an extremal hypergraph problem of Brown, Erdős, and Sós. Combinatorica , 26(6):627–645, 2006. doi:10.1007/s 00493-006-0035-9 . · doi ↗
2[2] T. Bohman and L. Warnke. Large girth approximate Steiner triple systems. J. London Math. Soc. , 2019. doi:10.1112/jlms.12242 . · doi ↗
3[3] W. G. Brown, P. Erdős, and V. T. Sós, Some extremal problems on r 𝑟 r -graphs. In New directions in the theory of graphs (Proc. Third Ann Arbor Conf., Univ. Michigan, Ann Arbor, Mich, 1971) (ed. F. Harary), pages 53–63. Academic Press, Cambridge, MA, 1973. https://old.renyi.hu/~p_erdos/1973-25.pdf
4[4] P. Erdős. Extremal problems in graph theory. In Theory of graphs and its applications (ed. M. Fiedler), pages 29–36. Publising House of the Czechoslovak Academy of Sciences, Prague, 1964. https://old.renyi.hu/~p_erdos/1964-06.pdf
5[5] P. Erdős. Problems and results in combinatorial analysis. In Colloquio Internazionale sulle Teorie Combinatorie (Rome, 1973) , Tomo II, pages 3–17. Atti dei Convegni Lincei , No. 17, Accad. Naz. Lincei, Rome 1976. https://old.renyi.hu/~p_erdos/1976-35.pdf
6[6] P. Erdős, P. Frankl, and V. Rödl. The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent. Graphs Combin. , 2(1):113–121, 1986. doi:10.1007/BF 01788085 . · doi ↗
7[7] G. Ge and C. Shangguan. Sparse hypergraphs: new bounds and constructions. ar Xiv:1706.03306 , 2017.
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TL;DR

Contribution

Findings

Abstract

Peer Reviews

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Approximate Steiner (r−1,r,n)(r-1,r,n)(r−1,r,n)-systems

Abstract

Theorem 1**.**

Proof.

Proposition 2**.**

Proof.

Proposition 3**.**

Proof.

Theorem 4**.**

Proof.

Corollary 5**.**

Approximate Steiner $(r-1,r,n)$ -systems

Theorem 1.

Proposition 2.

Proposition 3.

Theorem 4.

Corollary 5.