Related papers: On the EKR Module property

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|…

A family of sets is intersecting if any two sets in the family intersect. Given a graph $G$ and an integer $r\geq 1$, let $\mathcal{I}^{(r)}(G)$ denote the family of independent sets of size $r$ of $G$. For a vertex $v$ of $G$, the family…

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…

In 1987, Frankl proved an influential stability result for the Erd\H os--Ko--Rado theorem, which bounds the size of an intersecting family in terms of its distance from the nearest (subset of) star or trivial intersecting family. It is a…

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…

A collection of sets is {\em intersecting} if every two members have nonempty intersection. We describe the structure of intersecting families of $r$-sets of an $n$-set whose size is quite a bit smaller than the maximum ${n-1 \choose r-1}$…

The main object of this paper is to determine the maximum number of $\{0,\pm 1\}$-vectors subject to the following condition. All vectors have length $n$, exactly $k$ of the coordinates are $+1$ and one is $-1$, $n \geq 2k$. Moreover, there…

We give a proof of the Erd\H{o}s-Ko-Rado Theorem using the Borel Fixed Point Theorem from algebraic group theory. This perspective gives a strong analogy between the Erd\H{o}s-Ko-Rado Theorem and (generalizations of) the Gerstenhaber…

Let $q$ be a power of a prime number and $V$ be the $2$-dimensional column vector space over a finite field $\mathbb{F}_{q}$. Assume that $SL_2(V)<G\leq GL_2(V)$. In this paper we prove an Erd{\H{o}}s-Ko-Rado theorem for intersecting sets…

We give simpler algebraic proofs of uniqueness for several Erd\H{o}s-Ko-Rado results, i.e., that the canonically intersecting families are the only largest intersecting families. Using these techniques, we characterize the largest partially…

A family of subsets $\mathcal{F}\subseteq {[n]\choose k}$ is called intersecting if any two of its members share a common element. Consider an intersecting family, a direct problem is to determine its maximal size and the inverse problem is…

This paper establishes an analog of the Erd\H{o}s-Ko-Rado theorem to polynomial rings over finite fields, affirmatively answering a conjecture of C. Tompkins. A $k$-uniform family of subsets of a set of finite size $n$ is $l$-intersecting…

Let $\A\subset\binom{[n]}{r}$ be a compressed, intersecting family and let $X\subset[n]$. Let $\A(X)={A\in\A:A\cap X\ne\emptyset}$ and $\S_{n,r}=\binom{[n]}{r}({1})$. Motivated by the Erd\H{o}s-Ko-Rado theorem, Borg asked for which…

In this paper we study a question related to the classical Erd\H{o}s-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…

This work addresses the existence of transitive extensions of certain infinite permutation groups which arise as the automorphism groups of model-theoretic structures which are generic in the Fra\"iss\'e sense. The study of transitive…

Let $S_n$ be the symmetric group on $n$ points. Deza and Frankl [M. Deza and P. Frankl, On the maximum number of permutations with given maximal or minimal distance, J. Combin. Theory Ser. A 22 (1977) 352--360] proved that if ${\cal F}$ is…

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…

In this paper we give a proof that the largest set of perfect matchings, in which any two contain a common edge, is the set of all perfect matchings that contain a fixed edge. This is a version of the famous Erd\H{o}s-Ko-Rado theorem for…

A $k$-uniform family of subsets of $[n]$ is intersecting if it does not contain a disjoint pair of sets. The study of intersecting families is central to extremal set theory, dating back to the seminal Erd\H{o}s-Ko-Rado theorem of 1961 that…

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…