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We study a variant of the Erd\H{o}s Matching Problem in random hypergraphs. Let $\mathcal{K}_p(n,k)$ denote the Erd\H{o}s-R\'enyi random $k$-uniform hypergraph on $n$ vertices where each possible edge is included with probability $p$. We show that when $n\gg k^{2}s$ and $p$ is not too small, with high probability, the maximum number of edges in a sub-hypergraph of $\mathcal{K}_p(n,k)$ with matching number $s$ is obtained by the trivial sub-hypergraphs, i.e. the sub-hypergraph consisting of all edges containing at least one vertex in a fixed set of $s$ vertices., Comment: 14 pages

We consider families, $\mathcal{F}$ of $k$-subsets of an $n$-set. For integers $r\geq 2$, $t\geq 1$, $\mathcal{F}$ is called $r$-wise $t$-intersecting if any $r$ of its members have at least $t$ elements in common. The most natural construction of such a family is the full $t$-star, consisting of all $k$-sets containing a fixed $t$-set. In the case $r=2$ the Exact Erd\H{o}s-Ko-Rado Theorem shows that the full $t$-star is largest if $n\geq (t+1)(k-t+1)$. In the present paper, we prove that for $n\geq (2.5t)^{1/(r-1)}(k-t)+k$, the full $t$-star is largest in case of $r\geq 3$. Examples show that the exponent $\frac{1}{r-1}$ is best possible. This represents a considerable improvement on a recent result of Balogh and Linz.

Babai and Frankl posed the ``odd cover problem" of finding the minimum cardinality of a collection of complete bipartite graphs such that every edge of the complete graph of order $n$ is covered an odd number of times. In a previous paper with O'Neill, some of the authors proved that this value is always $\lceil n / 2 \rceil$ or $\lceil n / 2 \rceil + 1$ and that it is the former whenever $n$ is a multiple of $8$. In this paper, we determine this value to be $\lceil n / 2 \rceil$ whenever $n$ is odd or equivalent to $18$ modulo $24$. We also further the study of odd covers of graphs which are not complete, wherein edges are covered an odd number of times and nonedges an even number of times by the complete bipartite graphs in the collection. Among various results on disjoint unions, we find the minimum cardinality of an odd cover of a union of odd cliques and of a union of cycles., Comment: 19 pages, 6 figures

Let us consider a collection $\mathcal G$ of codewords of length $n$ over an alphabet of size $s$. Let $t_1,\ldots, t_s$ be nonnegative integers. What is the maximum of $|\mathcal G|$ subject to the condition that any two codewords should have at least $t_i$ positions where both have letter $i$ ($1\le i\le s$). In the case $s=2$ it is a longstanding open question. Quite surprisingly, we obtain an almost complete answer for $s\ge 3$. The main tool is a correlation inequality.

A $k$-uniform family $\mathcal{F}$ is called intersecting if $F\cap F'\neq \emptyset$ for all $F,F'\in \mathcal{F}$. The shadow family $\partial \mathcal{F}$ is the family of $(k-1)$-element sets that are contained in some members of $\mathcal{F}$. The shadow degree (or minimum positive co-degree) of $\mathcal{F}$ is defined as the maximum integer $r$ such that every $E\in \partial \mathcal{F}$ is contained in at least $r$ members of $\mathcal{F}$. In 2021, Balogh, Lemons and Palmer determined the maximum size of an intersecting $k$-uniform family with shadow degree at least $r$ for $n\geq n_0(k,r)$, where $n_0(k,r)$ is doubly exponential in $k$ for $4\leq r\leq k$. In the present paper, we present a short proof of this result for $n\geq 2(r+1)^rk \frac{\binom{2k-1}{k}}{\binom{2r-1}{r}}$ and $4\leq r\leq k$.

There are four non-isomorphic configurations of triples that can form a triangle in a $3$-uniform hypergraph. Forbidding different combinations of these four configurations, fifteen extremal problems can be defined, several of which already appeared in the literature in some different context. Here we systematically study all of these problems solving the new cases exactly or asymptotically. In many cases we also characterize the extremal constructions., Comment: 44 pages, 4 figures

A family of sets is $r$-wise agreeing if for any $r$ sets from the family there is an element $x$ that is either contained in all or contained in none of the $r$ sets. The study of such families is motivated by questions in discrete optimization. In this paper, we determine the size of the largest non-trivial $r$-wise agreeing family. This can be seen as a generalization of the classical Brace-Daykin theorem.

Extending the notion of sunflowers, we call a family of at least two sets an odd-sunflower if every element of the underlying set is contained in an odd number of sets or in none of them. It follows from the Erd\H os--Szemer\'edi conjecture, recently proved by Naslund and Sawin, that there is a constant $\mu<2$ such that every family of subsets of an $n$-element set that contains no odd-sunflower consists of at most $\mu^n$ sets. We construct such families of size at least $1.5021^n$. We also characterize minimal odd-sunflowers of triples.

Let $\mathcal{F}\subset \binom{X}{k}$ be a family consisting of $k$-subsets of the $n$-set $X$. Suppose that $\mathcal{F}$ is intersecting, i.e., $F\cap F'\neq \emptyset$ for all $F,F'\in \mathcal{F}$. Let $\Delta(\mathcal{F})$ be the maximum degree of $\mathcal{F}$. For a constant $C\geq 1$ the $C$-diversity, $\gamma_C(\mathcal{F})$ is defined as $|\mathcal{F}|-C\Delta(\mathcal{F})$. Define $\mathcal{F}_{123} =\left\{F\in \binom{X}{k}\colon |F\cap \{1,2,3\}|=2\right\}$. It has $C$-diversity $(3-2C)\binom{n-3}{k-2}$. The main result shows that for $1< C<\frac{3}{2}$ and $n\geq \frac{42}{3-2C}k$, $\gamma_C(\mathcal{F})\leq \gamma_C(\mathcal{F}_{123})$ with equality if and only if $\mathcal{F}$ is isomorphic to $\mathcal{F}_{123}$. For the case of ordinary diversity $(C=1)$ a strong stability is proven., Comment: 17 pages

Let $X$ be an $n$-element set. A set-pair system $\mbox{$\cal P$}=\{(A_i,B_i)\}_{1\leq i\leq m}$ is a collection of pairs of disjoint subsets of $X$. It is called skew Bollob\'as system if $A_i\cap B_j\neq \emptyset$ for all $1\leq i

A family $\mathcal{F}\subset \binom{[n]}{k}$ is called an intersecting family if $F\cap F'\neq \emptyset$ for all $F,F'\in \mathcal{F}$. If $\cap \mathcal{F}\neq \emptyset$ then $\mathcal{F}$ is called a star. The diversity of an intersecting family $\mathcal{F}$ is defined as the minimum number of $k$-sets in $\mathcal{F}$, whose deletion results in a star. In the present paper, we prove that for $n>36k$ any intersecting family $\mathcal{F}\subset \binom{[n]}{k}$ has diversity at most $\binom{n-3}{k-2}$, which improves the previous best bound $n>72k$ due to the first author. This result is derived from some strong bounds concerning the maximum degree of large intersecting families. Some related results are established as well., Comment: arXiv admin note: substantial text overlap with arXiv:2207.05487

An $r$-uniform hypergraph has $(q,p)$-property if any set of $q$ vertices spans a complete sub-hypergraph on $p$ vertices. Let $t_r(n,q,p)$ be the minimum edge density of an $n$-vertex $r$-uniform hypergraph with {\em $(q,p)$-property} and let $t_r(q,p)=\lim_{n\to\infty}t_r(n,q,p)$. A disjoint union of $k$ complete hypergraphs has $(q,\lceil q/k\rceil)$-property, which gives $t_r((q,\lceil{q/k}\rceil))\le 1/k^{r-1}$. The first author, Huang and R\"odl showed that these constructions are the best asymptotically, that is, $\lim_{q\to\infty}t_r((q,\lceil{q/k}\rceil))=1/k^{r-1}$. They asked whether it is true for all real number $\gamma\ge1$ that $\lim_{q\to\infty}t_r((q,\lceil{q/\gamma}\rceil))=1/\lfloor{\gamma}\rfloor^{r-1}$. In this paper, we give positive answers to this question for a small range of real numbers, and, on the other hand, provide new constructions that give negative answers for many other ranges., Comment: 12 pages

Let $[n]=X_1\cup X_2\cup X_3$ be a partition with $\lfloor\frac{n}{3}\rfloor \leq |X_i|\leq \lceil\frac{n}{3}\rceil$ and define $\mathcal{G}=\{G\subset [n]\colon |G\cap X_i|\leq 1, 1\leq i\leq 3\}$. It is easy to check that the trace $\mathcal{G}_{\mid Y}:=\{G\cap Y\colon G\in \mathcal{G}\}$ satisfies $|\mathcal{G}_{\mid Y}|\leq 12$ for all 4-sets $Y\subset [n]$. For $n\geq 25$ it is proven that whenever $\mathcal{F}\subset 2^{[n]}$ satisfies $|\mathcal{F}|>|\mathcal{G}|$ then $|\mathcal{F}_{\mid C}|\geq 13$ for some $C\subset [n]$, $|C|=4$. Several further results of a similar flavor are established as well.

For a family $\mathcal{F}\subset \binom{[n]}{k}$ and two elements $x,y\in [n]$ define $\mathcal{F}(\bar{x},\bar{y})=\{F\in \mathcal{F}\colon x\notin F,\ y\notin F\}$. The double-diversity $\gamma_2(\mathcal{F})$ is defined as the minimum of $|\mathcal{F}(\bar{x},\bar{y})|$ over all pairs $x,y$. Let $\mathcal{L}\subset\binom{[7]}{3}$ consist of the seven lines of the Fano plane. For $n\geq 7$, $k\geq 3$ one defines the Fano $k$-graph $\mathcal{F}_{\mathcal{L}}$ as the collection of all $k$-subsets of $[n]$ that contain at least one line. It is proven that for $n\geq 13k^2$ the Fano $k$-graph is the essentially unique family maximizing the double diversity over all $k$-graphs without a pair of disjoint edges. Some similar, although less exact results are proven for triple and higher diversity as well.

We determine the maximum possible number of edges of a graph with $n$ vertices, matching number at most $s$ and clique number at most $k$ for all admissible values of the parameters.

For positive integers $n>k>t$ let $\binom{[n]}{k}$ denote the collection of all $k$-subsets of the standard $n$-element set $[n]=\{1,\ldots,n\}$. Subsets of $\binom{[n]}{k}$ are called $k$-graphs. A $k$-graph $\mathcal{F}$ is called $t$-intersecting if $|F\cap F'|\geq t$ for all $F,F'\in \mathcal{F}$. One of the central results of extremal set theory is the Erd\H{o}s-Ko-Rado Theorem which states that for $n\geq (k-t+1)(t+1)$ no $t$-intersecting $k$-graph has more than $\binom{n-t}{k-t}$ edges. For $n$ greater than this threshold the $t$-star (all $k$-sets containing a fixed $t$-set) is the only family attaining this bound. Define $\mathcal{F}(i)=\{F\setminus \{i\}\colon i\in F\in \mathcal{F}\}$. The quantity $\varrho(\mathcal{F})=\max\limits_{1\leq i\leq n}|\mathcal{F}(i)|/|\mathcal{F}|$ measures how close a $k$-graph is to a star. The main result (Theorem 1.5) shows that $\varrho(\mathcal{F})>1/d$ holds if $\mathcal{F}$ is 1-intersecting, $|\mathcal{F}|>2^dd^{2d+1}\binom{n-d-1}{k-d-1}$ and $n\geq 4(d-1)dk$. Such a statement can be deduced from the results of \cite{F78-2} and \cite{DF}, however only for much larger values of $n/k$ and/or $n$. The proof is purely combinatorial, it is based on a new method: shifting ad extremis. The same method is applied to obtain some nearly optimal bounds in the case of $t\geq 2$ (Theorem 1.11) along with a number of related results., Comment: 35 pages

We consider families of k-subsets of the standard n-set. Two families F, G are said to be cross-intersecting if every member of F has non-empty intersection with every member of G. A family is called non-trivial if the intersection of all its members is empty. Supposing that F and G are non-trivial and cross-intersecting, we determine the maximum of |F|+|G|. For the proof a strengthened version of the so-called shifting technique is introduced. The corresponding problem for families of different uniformities is solved as well., Comment: 23 pages

We consider $k$-graphs on $n$ vertices, that is, $\mathcal{F}\subset \binom{[n]}{k}$. A $k$-graph $\mathcal{F}$ is called intersecting if $F\cap F'\neq \emptyset$ for all $F,F'\in \mathcal{F}$. In the present paper we prove that for $k\geq 7$, $n\geq 2k$, any intersecting $k$-graph $\mathcal{F}$ with covering number at least three, satisfies $|\mathcal{F}|\leq \binom{n-1}{k-1}-\binom{n-k}{k-1}-\binom{n-k-1}{k-1}+\binom{n-2k}{k-1}+\binom{n-k-2}{k-3}+3$, the best possible upper bound which was proved in \cite{F80} subject to exponential constraints $n>n_0(k)$., Comment: 30 pages

Given a graph $G$, let $\mu(G)$ denote the size of the smallest maximal independent set in $G$. A family of subsets is called a star if some element is in every set of the family. A split vertex has degree at least 3. Holroyd and Talbot conjectured the following Erd\H{o}s-Ko-Rado type statement about intersecting families of independent sets in graphs: if $1\le r\le \mu(G)/2$ then there is an intersecting family of independent $r$-sets of maximum size that is a star. In this paper we prove similar statements for sparse graphs on $n$ vertices: roughly, for graphs of bounded average degree with $r\le O(n^{1/3})$, for graphs of bounded degree with $r\le O(n^{1/2})$, and for trees having a bounded number of split vertices with $r\le O(n^{1/2})$., Comment: Correction of typos and inclusion of additional history

Two families $\mathcal{F},\mathcal{G}$ of $k$-subsets of $\{1,2,\ldots,n\}$ are called non-trivial cross-intersecting if $F\cap G\neq \emptyset$ for all $F\in \mathcal{F}, G\in \mathcal{G}$ and $\cap \{F\colon F\in \mathcal{F}\}=\emptyset=\cap \{G\colon G\in \mathcal{G}\}$. In the present paper, we determine the maximum product of the sizes of two non-trivial cross-intersecting families of $k$-subsets of $\{1,2,\ldots,n\}$ for $n\geq 4k$, $k\geq 8$, which is a product version of the classical Hilton-Milner Theorem., Comment: 24 pages

Two families $\mathcal{F},\mathcal{G}$ of $k$-subsets of $\{1,2,\ldots,n\}$ are called non-trivial cross $t$-intersecting if $|F\cap G|\geq t$ for all $F\in \mathcal{F}, G\in \mathcal{G}$ and $|\cap \{F\colon F\in \mathcal{F}\}|

Let $\mathcal{F},\mathcal{G}$ be two cross-intersecting families of $k$-subsets of $\{1,2,\ldots,n\}$. Let $\mathcal{F}\wedge \mathcal{G}$, $\mathcal{I}(\mathcal{F},\mathcal{G})$ denote the families of all intersections $F\cap G$ with $F\in \mathcal{F},G\in \mathcal{G}$, and all distinct intersections $F\cap G$ with $F\neq G, F\in \mathcal{F},G\in \mathcal{G}$, respectively. For a fixed $T\subset \{1,2,\ldots,n\}$, let $\mathcal{S}_T$ be the family of all $k$-subsets of $\{1,2,\ldots,n\}$ containing $T$. In the present paper, we show that $|\mathcal{F}\wedge \mathcal{G}|$ is maximized when $\mathcal{F}=\mathcal{G}=\mathcal{S}_{\{1\}}$ for $n\geq 2k^2+8k$, while surprisingly $|\mathcal{I}(\mathcal{F}, \mathcal{G})|$ is maximized when $\mathcal{F}=\mathcal{S}_{\{1,2\}}\cup \mathcal{S}_{\{3,4\}}\cup \mathcal{S}_{\{1,4,5\}}\cup \mathcal{S}_{\{2,3,6\}}$ and $\mathcal{G}=\mathcal{S}_{\{1,3\}}\cup \mathcal{S}_{\{2,4\}}\cup \mathcal{S}_{\{1,4,6\}}\cup \mathcal{S}_{\{2,3,5\}}$ for $n\geq 100k^2$. The maximum number of distinct intersections in a $t$-intersecting family is determined for $n\geq 3(t+2)^3k^2$ as well.

We say that $k$ graphs $G_1,G_2,\dots,G_k$ on a common vertex set of size $n$ contain a rainbow copy of a graph $H$ if their union contains a copy of $H$ with each edge belonging to a distinct $G_i$. We provide a counterexample to a conjecture of Frankl on the maximum product of the sizes of the edge sets of three graphs avoiding a rainbow triangle. We propose an alternative conjecture, which we prove under the additional assumption that the union of the three graphs is complete. Furthermore, we determine the maximum product of the sizes of the edge sets of three graphs or four graphs avoiding a rainbow path of length three., Comment: The paper has been expanded to include further results on paths

Let F,G,H be three graphs on the same n vertices. We consider the maximum of the sum and product of the number of their edges subject to the condition in the title.

In this paper, we show that, given two down-sets (simplicial complexes) there is a matching between them that matches disjoint sets and covers the smaller of the two down-sets. This result generalizes an unpublished result of Berge from circa 1980. The result has nice corollaries for cross-intersecting families and Chv\'atal's conjecture. More concretely, we show that Chv\'atal's conjecture is true for intersecting families with covering number $2$. A family $\mathcal F\subset 2^{[n]}$ is intersection-union (IU) if for any $A,B\in\mathcal F$ we have $1\le |A\cap B|\le n-1$. Using the aforementioned result, we derive several exact product- and sum-type results for IU-families.

We consider the minimum number of lines $h_n$ and $p_n$ needed to intersect or pierce, respectively, all the cells of the $n \times n$ chessboard. Determining these values can also be interpreted as a strengthening of the classical plank problem for integer points. Using the symmetric plank theorem of K. Ball, we prove that $h_n = \lceil \frac n 2 \rceil$ for each $n \geq 1$. Studying the piercing problem, we show that $0.7n \leq p_n \leq n-1$ for $n\geq 3$, where the upper bound is conjectured to be sharp. The lower bound is proven by using the linear programming method, whose limitations are also demonstrated., Comment: 15 pages, 7 figures. Final, accepted version. Color of figures modified in order to comply with BW print

We introduce the following variant of the VC-dimension. Given $S \subseteq \{0, 1\}^n$ and a positive integer $d$, we define $\mathbb{U}_d(S)$ to be the size of the largest subset $I \subseteq [n]$ such that the projection of $S$ on every subset of $I$ of size $d$ is the $d$-dimensional cube. We show that determining the largest cardinality of a set with a given $\mathbb{U}_d$ dimension is equivalent to a Tur\'an-type problem related to the total number of cliques in a $d$-uniform hypergraph. This allows us to beat the Sauer--Shelah lemma for this notion of dimension. We use this to obtain several results on $\Sigma_3^k$-circuits, i.e., depth-$3$ circuits with top gate OR and bottom fan-in at most $k$: * Tight relationship between the number of satisfying assignments of a $2$-CNF and the dimension of the largest projection accepted by it, thus improving Paturi, Saks, and Zane (Comput. Complex. '00). * Improved $\Sigma_3^3$-circuit lower bounds for affine dispersers for sublinear dimension. Moreover, we pose a purely hypergraph-theoretic conjecture under which we get further improvement. * We make progress towards settling the $\Sigma_3^2$ complexity of the inner product function and all degree-$2$ polynomials over $\mathbb{F}_2$ in general. The question of determining the $\Sigma_3^3$ complexity of IP was recently posed by Golovnev, Kulikov, and Williams (ITCS'21).

For $n > 2k \geq 4$ we consider intersecting families $\mathcal F$ consisting of $k$-subsets of $\{1, 2, \ldots, n\}$. Let $\mathcal I(\mathcal F)$ denote the family of all distinct intersections $F \cap F'$, $F \neq F'$ and $F, F'\in \mathcal F$. Let $\mathcal A$ consist of the $k$-sets $A$ satisfying $|A \cap \{1, 2, 3\}| \geq 2$. We prove that for $n \geq 50 k^2$ $|\mathcal I(\mathcal F)|$ is maximized by $\mathcal A$.

A family $\mathcal F$ has covering number $\tau$ if the size of the smallest set intersecting all sets from $\mathcal F$ is equal to $\tau$. Let $M(n,k,\tau)$ stand for the size of the largest intersecting family $\mathcal F$ of $k$-element subsets of $\{1,\ldots,n\}$ with covering number $\tau$. It is a classical result of Erd\H os and Lov\'asz that $M(n,k,k)\le k^k$ for any $n$. In this short note, we explore the behaviour of $M(n,k,\tau)$ for $nk-\frac 12k^{1/2}$.

For a family $\mathcal F$, let $\mathcal D(\mathcal F)$ stand for the family of all sets that can be expressed as $F\setminus G$, where $F,G\in \mathcal F$. A family $\mathcal F$ is intersecting if any two sets from the family have non-empty intersection. In this paper, we study the following question: what is the maximum of $|\mathcal D(\mathcal F)|$ for an intersecting family of $k$-element sets? Frankl conjectured that the maximum is attained when $\mathcal F$ is the family of all sets containing a fixed element. We show that this holds if $n \ge 50k\ln k$ and $k \ge 50$. At the same time, we provide a counterexample for $n< 4k$.

Let $\mathcal{A}_1,\ldots,\mathcal{A}_m$ be families of $k$-subsets of an $n$-set. Suppose that one cannot choose pairwise disjoint edges from $s+1$ distinct families. Subject to this condition we investigate the maximum of $|\mathcal{A}_1|+\ldots+|\mathcal{A}_m|$. Note that the subcase $m=s+1$, $\mathcal{A}_1=\ldots=\mathcal{A}_m$ is the Erd\H{o}s Matching Conjecture, one of the most important open problems in extremal set theory. We provide some upper bounds, a general conjecture and its solution for the range $n\geq 4k^2s$., Comment: 10 pages

In a recent breakthrough, Adiprasito, Avvakumov, and Karasev constructed a triangulation of the $n$-dimensional real projective space with a subexponential number of vertices. They reduced the problem to finding a small downward closed set-system $\cal F$ covering an $n$-element ground set which satisfies the following condition: For any two disjoint members $A, B\in\cal F$, there exist $a\in A$ and $b\in B$ such that either $B\cup\{a\}\in\cal F$ and $A\cup\{b\}\setminus\{a\}\in\cal F$, or $A\cup\{b\}\in\cal F$ and $B\cup\{a\}\setminus\{b\}\in\cal F$. Denoting by $f(n)$ the smallest cardinality of such a family $\cal F$, they proved that $f(n)<2^{O(\sqrt{n}\log n)}$, and they asked for a nontrivial lower bound. It turns out that the construction of Adiprasito et al. is not far from optimal; we show that $2^{(1.42+o(1))\sqrt{n}}\le f(n)\le 2^{(1+o(1))\sqrt{2n\log n}}$. We also study a variant of the above problem, where the condition is strengthened by also requiring that for any two disjoint members $A, B\in\cal F$ with $|A|>|B|$, there exists $a\in A$ such that $B\cup\{a\}\in\cal F$. In this case, we prove that the size of the smallest $\cal F$ satisfying this stronger condition lies between $2^{\Omega(\sqrt{n}\log n)}$ and $2^{O(n\log\log n/\log n)}$.

Given $n$ pairwise disjoint sets $X_1,\ldots, X_n$, we call the elements of $S=X_1\times\ldots\times X_n$ strings. A nonempty set of strings $W\subseteq S$ is said to be well-connected if for every $v\in W$ and for every $i\, (1\le i\le n)$, there is another element $v'\in W$ which differs from $v$ only in its $i$th coordinate. We prove a conjecture of Yaokun Wu and Yanzhen Xiong by showing that every set of more than $\prod_{i=1}^n|X_i|-\prod_{i=1}^n(|X_i|-1)$ strings has a well-connected subset. This bound is tight., Comment: 6 pages

For a $k$-graph $\mathcal{F}\subset \binom{[n]}{k}$, the clique number of $\mathcal{F}$ is defined to be the maximum size of a subset $Q$ of $[n]$ with $\binom{Q}{k}\subset \mathcal{F}$. In the present paper, we determine the maximum number of edges in a $k$-graph on $[n]$ with matching number at most $s$ and clique number at least $q$ for $n\geq 8k^2s$ and for $q \geq (s+1)k-l$, $n\leq (s+1)k+s/(3k)-l$. Two special cases that $q=(s+1)k-2$ and $k=2$ are solved completely., Comment: 26 pages

Several new applications of Katona's circle are given., Comment: A paper to celebrate Katona's 80th birthday, 32 pages

Given a family of sets on the plane, we say that the family is intersecting if for any two sets from the family their interiors intersect. In this paper, we study intersecting families of triangles with vertices in a given set of points. In particular, we show that if a set $P$ of $n$ points is in convex position, then the largest intersecting family of triangles with vertices in $P$ contains at most $(\frac{1}{4}+o(1))\binom{n}{3}$ triangles.

Let $2^{[n]}$ and $\binom{[n]}{i}$ be the power set and the class of all $i$-subsets of $\{1,2,\cdots,n\}$, respectively. We call two families $\mathscr{A}$ and $\mathscr{B}$ cross-intersecting if $A\cap B\neq \emptyset$ for any $A\in \mathscr{A}$ and $B\in \mathscr{B}$. In this paper we show that, for $n\geq k+l,l\geq r\geq 1,c>0$ and $\mathscr{A}\subseteq \binom{[n]}{k},\mathscr{B}\subseteq \binom{[n]}{l}$, if $\mathscr{A}$ and $\mathscr{B}$ are cross-intersecting and $\binom{n-r}{l-r}\leq|\mathscr{B}|\leq \binom{n-1}{l-1}$, then $$|\mathscr{A}|+c|\mathscr{B}|\leq \max\left\{\binom{n}{k}-\binom{n-r}{k}+c\binom{n-r}{l-r},\ \binom{n-1}{k-1}+c\binom{n-1}{l-1}\right\}$$ and the families $\mathscr{A}$ and $\mathscr{B}$ attaining the upper bound are also characterized. This generalizes the corresponding result of Hilton and Milner for $c=1$ and $r=k=l$, and implies a result of Tokushige and the second author (Theorem 1.3)., Comment: 12 pages

Let $n>s>0$ be integers, $X$ an $n$-element set and $\mathscr{A}, \mathscr{B}\subset 2^X$ two families. If $|A\cup B|\le s$ for all $A\in\mathscr{A}, B\in \mathscr{B}$, then $\mathscr{A}$ and $\mathscr{B}$ are called cross $s$-union. Assuming that neither $\mathscr{A}$ nor $\mathscr{B}$ is empty, we prove several best possible bounds. In particular, we show that $|\mathscr{A}|+|\mathscr{B}|\le 1+\sum\limits_{0\le i\le s}{{n}\choose{i}}$. Supposing $n\ge 2s$ and $\mathscr{A},\mathscr{B}$ are antichains, we show that $|\mathscr{A}|+|\mathscr{B}|\le {{n}\choose{1}}+{{n}\choose{s-1}}$ unless $\mathscr{A}=\{\emptyset\}$ or $\mathscr{B}=\{\emptyset\}$. An analogous result for three families is established as well., Comment: 14 pages

Let $X$ be an $n$-element set, where $n$ is even. We refute a conjecture of J. Gordon and Y. Teplitskaya, according to which, for every maximal intersecting family $\mathcal{F}$ of $\frac{n}2$-element subsets of $X$, one can partition $X$ into $\frac{n}2$ disjoint pairs in such a way that no matter how we pick one element from each of the first $\frac{n}2 - 1$ pairs, the set formed by them can always be completed to a member of $\mathcal{F}$ by adding an element of the last pair. The above problem is related to classical questions in extremal set theory. For any $t\ge 2$, we call a family of sets $\mathcal{F}\subset 2^X$ {\em $t$-separable} if for any ordered pair of elements $(x,y)$ of $X$, there exists $F\in\mathcal{F}$ such that $F\cap\{x,y\}=\{x\}$. For a fixed $t, 2\le t\le 5$ and $n\rightarrow\infty$, we establish asymptotically tight estimates for the smallest integer $s=s(n,t)$ such that every family $\mathcal{F}$ with $|\mathcal{F}|\ge s$ is $t$-separable., Comment: 12 pages

It is well-known that a line can intersect at most $2n-1$ cells of the $n \times n$ chessboard. Here we consider the high dimensional version: how many cells of the $d$-dimensional $n\times \ldots \times n$ box can a hyperplane intersect? We also prove the lattice analogue of the following well-known fact. If $K,L$ are convex bodies in $R^d$ and $K\subset L$, then the surface area of $K$ is smaller than that of $L$.

Since its formulation, Tur\'an's hypergraph problems have been among the most challenging open problems in extremal combinatorics. One of them is the following: given a $3$-uniform hypergraph $\mathcal{F}$ on $n$ vertices in which any five vertices span at least one edge, prove that $|\mathcal{F}| \ge (1/4 -o(1))\binom{n}{3}$. The construction showing that this bound would be best possible is simply $\binom{X}{3} \cup \binom{Y}{3}$ where $X$ and $Y$ evenly partition the vertex set. This construction has the following more general $(2p+1, p+1)$-property: any set of $2p+1$ vertices spans a complete sub-hypergraph on $p+1$ vertices. One of our main results says that, quite surprisingly, for all $p>2$ the $(2p+1,p+1)$-property implies the conjectured lower bound.

In this paper, we study the maximum degree in non-empty induced subgraphs of the Kneser graph $KG(n,k)$. One of the main results asserts that, for $k>k_0$ and $n>64k^2$, whenever a non-empty subgraph has $m\ge k{n-2\choose k-2}$ vertices, its maximum degree is at least $\frac 12(1-\frac {k^2}n) m - {n-2\choose k-2}\ge 0.49 m$. This bound is essentially best possible. One of the intermediate steps is to obtain structural results on non-empty subgraphs with small maximum degree.

In this paper, we investigate Erd\H os--Ko--Rado type theorems for families of vectors from $\{0,\pm 1\}^n$ with fixed numbers of $+1$'s and $-1$'s. Scalar product plays the role of intersection size. In particular, we sharpen our earlier result on the largest size of a family of such vectors that avoids the smallest possible scalar product. We also obtain an exact result for the largest size of a family with no negative scalar products.