Related papers: Intersection Problems in Extremal Combinatorics: T…

We use an algebraic method to prove a degree version of the celebrated Erd\H os-Ko-Rado theorem: given $n>2k$, every intersecting $k$-uniform hypergraph $H$ on $n$ vertices contains a vertex that lies on at most $\binom{n-2}{k-2}$ edges.…

An ordered hypergraph is a hypergraph whose vertex set is linearly ordered, and a convex geometric hypergraph is a hypergraph whose vertex set is cyclically ordered. Extremal problems for ordered and convex geometric graphs have a rich…

We consider an Erd\H{o}s-Ko-Rado type sum that weights each member of a uniform family according to its smallest intersection with the rest of the family. We prove that once the ground set is sufficiently large this sum is at most one, with…

Consider a graph $G$ and a $k$-uniform hypergraph $\mathcal{H}$ on common vertex set $[n]$. We say that $\mathcal{H}$ is $G$-intersecting if for every pair of edges in $X,Y \in \mathcal{H}$ there are vertices $x \in X$ and $y \in Y$ such…

Let $k$ and $m$ be positive integers. A collection of $k$-multisets from $\{1,..., m \}$ is intersecting if every pair of multisets from the collection is intersecting. We prove that for $m \geq k+1$, the size of the largest such collection…

The Erd\H{o}s-Ko-Rado theorem states that for $r \leq \frac{n}{2}$, the largest intersecting family of $r$-subsets of $[n]$ is given by fixing a common element in all subsets, which trivially ensures pairwise intersection. We investigate…

A family $\mathcal{F}$ is $t$-$\it{intersecting}$ if any two members have at least $t$ common elements. Erd\H os, Ko, and Rado proved that the maximum size of a $t$-intersecting family of subsets of size $k$ is equal to $ {{n-t} \choose…

Let $\mathcal F\subset 2^{[n]}$ be an $s$-uniform family such that every two distinct sets have a nonempty intersection but intersect in at most $k$ elements. By the well-known Ray-Chaudhuri--Wilson theorem, since the intersections can take…

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…

A $(k,\ell)$-partition is a set partition which has $\ell$ blocks each of size $k$. Two uniform set partitions $P$ and $Q$ are said to be partially $t$-intersecting if there exist blocks $P_{i}$ in $P$ and $Q_{j}$ in $Q$ such that $\left|…

We study intersecting families of words from the Erd\H{o}s-Ko-Rado perspective. When the alphabet size is $2$, a maximum intersecting family is not necessarily a star. However, we prove that every maximum $3$-wise intersecting family is a…

A family of sets is intersecting if no two of its members are disjoint, and has the Erd\H{o}s-Ko-Rado property (or is EKR) if each of its largest intersecting subfamilies has nonempty intersection. Denote by $\mathcal{H}_k(n,p)$ the random…

Let $m$, $n$, and $k$ be integers satisfying $0 < k \leq n < 2k \leq m$. A family of sets $\mathcal{F}$ is called an $(m,n,k)$-intersecting family if $\binom{[n]}{k} \subseteq \mathcal{F} \subseteq \binom{[m]}{k}$ and any pair of members of…

The $t$-intersecting Erd\H{o}s-Ko-Rado theorem is the following statement: if $\mathcal{F} \subset \binom{[n]}{k}$ is a $t$-intersecting family of sets and $n\ge (t+1)(k-t+1)$, then $|\mathcal{F}| \le \binom{n-t}{k-t}$. The first proof of…

Two families of sets $\mathcal{A}$ and $\mathcal{B}$ are called cross-$t$-intersecting if $|A\cap B|\ge t$ for all $A\in \mathcal{A}$, $B\in \mathcal{B}$. An active problem in extremal set theory is to determine the maximum product of sizes…

We introduce a measure for subspaces of a vector space over a $q$-element field, and propose some extremal problems for intersecting families. These are $q$-analogue of Erd\H{o}s-Ko-Rado type problems, and we answer some of the basic…

A set partition is $c$-uniform if every block has size $c$. Two families of $c$-uniform partitions of a finite set are said to be cross $t$-intersecting if two partitions from different families share at least $t$ blocks. In this paper, we…

Many classical theorems in combinatorics establish the emergence of substructures within sufficiently large collections of objects. Well-known examples are Ramsey's theorem on monochromatic subgraphs and the Erd\H{o}s-Rado sunflower lemma.…

In 1965 Erd\H{o}s conjectured that the number of edges in k-uniform hypergraphs on n vertices in which the largest matching has s edges is maximized for hypergraphs of one of two special types. We settled this conjecture in the affirmative…

An intersecting family of sets is trivial if all of its members share a common element. Hilton and Milner proved a strong stability result for the celebrated Erd\H{o}s--Ko--Rado theorem: when $n> 2k$, every non-trivial intersecting family…