Distance graphs with large chromatic number and arbitrary girth

In this article we consider a problem related to two famous combinatorial topics. One of them concerns the chromatic number of the space. The other deals with graphs having big girth (the length of the shortest cycle) and large chromatic number. Namely, we prove that for any $l\in \mathbb{N}$ there exists a sequence of distance graphs in $\mathbb{R}^n$ with girth at least $l$ and the chromatic number equal to $(c+\bar{o}(1))^n$ with $c>1$.Comment: 8 page

Diameter graphs in R4\mathbb R^4

A \textit{diameter graph in $\mathbb R^d$} is a graph, whose set of vertices is a finite subset of $\mathbb R^d$ and whose set of edges is formed by pairs of vertices that are at diameter apart. This paper is devoted to the study of different extremal properties of diameter graphs in $\mathbb R^4$ and on a three-dimensional sphere. We prove an analogue of V\'azsonyi's and Borsuk's conjecture for diameter graphs on a three-dimensional sphere with radius greater than $1/\sqrt 2$. We prove Schur's conjecture for diameter graphs in $\mathbb R^4.$ We also establish the maximum number of triangles a diameter graph in $\mathbb R^4$ can have, showing that the extremum is attained only on specific Lenz configurations.Comment: 17 page

Structure and properties of large intersecting families

We say that a family of $k$-subsets of an $n$-element set is intersecting, if any two of its sets intersect. In this paper we study different extremal properties of intersecting families, as well as the structure of large intersecting families. We also give some results on $k$-uniform families without $s$ pairwise disjoint sets, related to the Erd\H{o}s Matching Conjecture. We prove a conclusive version of Frankl's theorem on intersecting families with bounded maximal degree. This theorem, along with its generalizations to cross-intersecting families, implies many results on the topic, obtained by Frankl, Frankl and Tokushige, Kupavskii and Zakharov and others. We study the structure of large intersecting families, obtaining some general structural theorems which generalize the results of Han and Kohayakawa, as well as Kostochka and Mubayi. We give degree and subset degree version of the Erd\H{o}s--Ko--Rado and the Hilton--Milner theorems, extending the results of Huang and Zhao, and Frankl, Han, Huang and Zhao. We also extend the range in which the degree version of the Erd\H{o}s Matching conjecture holds.Comment: This is a preliminary version of the text, which I decided to keep because other papers refer to the problems I posed in this version. By now, it was split into two papers: arXiv:1810.00915 and arXiv:1810.00920 . The presentation was improved and the results concerning the structure of intersecting families were generalized and strengthene

On random subgraphs of Kneser and Schrijver graphs

A Kneser graph $KG_{n,k}$ is a graph whose vertices are in one-to-one correspondence with $k$-element subsets of $[n],$ with two vertices connected if and only if the corresponding sets do not intersect. A famous result due to Lov\'asz states that the chromatic number of a Kneser graph $KG_{n,k}$ is equal to $n-2k+2$. In this paper we study the chromatic number of a random subgraph of a Kneser graph $KG_{n,k}$ as $n$ grows. A random subgraph $KG_{n,k}(p)$ is obtained by including each edge of $KG_{n,k}$ with probability $p$. For a wide range of parameters $k = k(n), p = p(n)$ we show that $\chi(KG_{n,k}(p))$ is very close to $\chi(KG_{n,k}),$ a.a.s. differing by at most 4 in many cases. Moreover, we obtain the same bounds on the chromatic numbers for the so-called Schrijver graphs, which are known to be vertex-critical induced subgraphs of Kneser graphs

Two problems on matchings in set families - in the footsteps of Erd\H{o}s and Kleitman

The families $\mathcal F_1,\ldots, \mathcal F_s\subset 2^{[n]}$ are called $q$-dependent if there are no pairwise disjoint $F_1\in \mathcal F_1,\ldots, F_s\in\mathcal F_s$ satisfying $|F_1\cup\ldots\cup F_s|\le q.$ We determine $\max |\mathcal F_1|+\ldots +|\mathcal F_s|$ for all values $n\ge q,s\ge 2$. The result provides a far-reaching generalization of an important classical result of Kleitman. The well-known Erd\H os Matching Conjecture suggests the largest size of a family $\mathcal F\subset {[n]\choose k}$ with no $s$ pairwise disjoint sets. After more than 50 years its full solution is still not in sight. In the present paper, we provide a Hilton-Milner-type stability theorem for the Erd\H{o}s Matching Conjecture in a relatively wide range, in particular, for $n\ge (2+o(1))sk$ with $o(1)$ depending on $s$ only. This is a considerable improvement of a classical result due to Bollob\'as, Daykin and Erd\H{o}s. We apply our results to advance in the following anti-Ramsey-type problem, proposed by \"Ozkahya and Young. Let $ar(n,k,s)$ be the minimum number $x$ of colors such that in any coloring of the $k$-element subsets of $[n]$ with $x$ (non-empty) colors there is a \textit{rainbow matching} of size $s$, that is, $s$ sets of different colors that are pairwise disjoint. We prove a stability result for the problem, which allows to determine $ar(n,k,s)$ for all $k\ge 3$ and $n\ge sk+(s-1)(k-1).$ Some other consequences of our results are presented as well

When are epsilon-nets small?

In many interesting situations the size of epsilon-nets depends only on $\epsilon$ together with different complexity measures. The aim of this paper is to give a systematic treatment of such complexity measures arising in Discrete and Computational Geometry and Statistical Learning, and to bridge the gap between the results appearing in these two fields. As a byproduct, we obtain several new upper bounds on the sizes of epsilon-nets that generalize/improve the best known general guarantees. In particular, our results work with regimes when small epsilon-nets of size $o(\frac{1}{\epsilon})$ exist, which are not usually covered by standard upper bounds. Inspired by results in Statistical Learning we also give a short proof of the Haussler's upper bound on packing numbers.Comment: 22 pages; minor changes, accepted versio

Sharp results concerning disjoint cross-intersecting families

For an $n$-element set $X$ let $\binom{X}{k}$ be the collection of all its $k$-subsets. Two families of sets $\mathcal A$ and $\mathcal B$ are called cross-intersecting if $A\cap B \neq \emptyset$ holds for all $A\in\mathcal A$, $B\in\mathcal B$. Let $f(n,k)$ denote the maximum of $\min\{|\mathcal A|, |\mathcal B|\}$ where the maximum is taken over all pairs of {\em disjoint}, cross-intersecting families $\mathcal A, \mathcal B\subset\binom{[n]}{k}$. Let $c=\log_2e$. We prove that $f(n,k)=\left\lfloor\frac12\binom{n-1}{k-1}\right\rfloor$ essentially iff $n>ck^2$ (cf. Theorem~1.4 for the exact statement). Let $f^*(n,k)$ denote the same maximum under the additional restriction that the intersection of all members of both $\mathcal A$ and $\mathcal B$ are empty. For $k\ge5$ and $n\ge k^3$ we show that $f^*(n,k)=\left\lfloor\frac12\left(\binom{n-1}{k-1}-\binom{n-2k}{k-1}\right)\right\rfloor+1$ and the restriction on $n$ is essentially sharp (cf. Theorem~5.4)

Intersection theorems for {0,±1}\{0,\pm 1\}-vectors and ss-cross-intersecting families

In this paper we study two directions of extending the classical Erd\H os-Ko-Rado theorem which states that any family of $k$-element subsets of the set $[n] = \{1,\ldots,n\}$ in which any two sets intersect, has cardinality at most ${n-1\choose k-1}$. In the first part of the paper we study the families of $\{0,\pm 1\}$-vectors. Denote by $\mathcal L_k$ the family of all vectors $\mathbf v$ from $\{0,\pm 1\}^n$ such that $\langle\mathbf v,\mathbf v\rangle = k$. For any $k$, most $l$ and sufficiently large $n$ we determine the maximal size of the family $\mathcal V\subset \mathcal L_k$ such that for any $\mathbf v,\mathbf w\in \mathcal V$ we have $\langle \mathbf v,\mathbf w\rangle\ge l$. We find some exact values of this function for all $n$ for small values of $k$. In the second part of the paper we study cross-intersecting pairs of families. We say that two families are $\mathcal A, \mathcal B$ are \textit{$s$-cross-intersecting}, if for any $A\in\mathcal A,B\in \mathcal B$ we have $|A\cap B|\ge s$. We also say that a set family $\mathcal A$ is {\it $t$-intersecting}, if for any $A_1,A_2\in \mathcal A$ we have $|A_1\cap A_2|\ge t$. For a pair of nonempty $s$-cross-intersecting $t$-intersecting families $\mathcal A,\mathcal B$ of $k$-sets, we determine the maximal value of $|\mathcal A|+|\mathcal B|$ for $n$ sufficiently large.Comment: This version contains a correction of an error, kindly pointed out to us by Danila Cherkashin and Sergei Kiselev. Notably, the statement of Theorem 5 part 2 is differen

Sharp bounds for the chromatic number of random Kneser graphs

Given positive integers $n\ge 2k$, the Kneser graph $KG_{n,k}$ is a graph whose vertex set is the collection of all $k$-element subsets of the set $\{1,\ldots, n\}$, with edges connecting pairs of disjoint sets. One of the classical results in combinatorics, conjectured by Kneser and proved by Lov\'asz, states that the chromatic number of $KG_{n,k}$ is equal to $n-2k+2$. In this paper, we study the chromatic number of the {\it random Kneser graph} $KG_{n,k}(p)$, that is, the graph obtained from $KG_{n,k}$ by including each of the edges of $KG_{n,k}$ independently and with probability $p$. We prove that, for any fixed $k\ge 3$, $\chi(KG_{n,k}(1/2)) = n-\Theta(\sqrt[2k-2]{\log_2 n})$, as well as $\chi(KG_{n,2}(1/2)) = n-\Theta(\sqrt[2]{\log_2 n \cdot \log_2\log_2 n})$. We also prove that, for any fixed $l\ge 6$ and $k\ge C\sqrt[2l-3]{\log n}$, we have $\chi(KG_{n,k}(1/2))\ge n-2k+2-2l$, where $C=C(l)$ is an absolute constant. This significantly improves previous results on the subject, obtained by Kupavskii and by Alishahi and Hajiabolhassan. We also discuss an interesting connection to an extremal problem on embeddability of complexes

Partition-free families of sets

Let $m(n)$ denote the maximum size of a family of subsets which does not contain two disjoint sets along with their union. In 1968 Kleitman proved that $m(n) = {n\choose m+1}+\ldots +{n\choose 2m+1}$ if $n=3m+1$. Confirming the conjecture of Kleitman, we establish the same equality for the cases $n=3m$ and $n=3m+2$, and also determine all extremal families. Unlike the case $n=3m+1$, the extremal families are not unique. This is a plausible reason behind the relative difficulty of our proofs. We completely settle the case of several families as well