Related papers: Vector sum-intersection theorems

The matching number of a family of subsets of an $n$-element set is the maximum number of pairwise disjoint sets. The families with matching number $1$ are called intersecting. The famous Erd\H os-Ko-Rado theorem determines the size of the…

A family F is intersecting if any two members have a nonempty intersection. Erdos, Ko, and Rado showed that |F|\leq {n-1\choose k-1} holds for an intersecting family of k-subsets of [n]:={1,2,3,...,n}, n\geq 2k. For n> 2k the only extremal…

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.…

The celebrated Erd\H{o}s-Ko-Rado theorem shows that for $n \ge 2k$ the largest intersecting $k$-uniform set family on $[n]$ has size $\binom{n-1}{k-1}$. It is natural to ask how far from intersecting larger set families must be. Katona,…

Two families of sets \(\mathcal{A}\) and \(\mathcal{B}\) are called \emph{cross-\(t\)-intersecting} if \(|A \cap B| \geq t\) for all \(A \in \mathcal{A}\) and \(B \in \mathcal{B}\). Determining the maximum product of sizes for such…

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…

A family $\mathcal{F}$ of subsets of a set $X$ is $t$-intersecting if $\vert A_i \cap A_j \vert \geq t$ for every $A_i, \; A_j \in \mathcal{F}$. We study intersecting families in the Hamming geometry. Given $X=\mathbb{F}_q^3$ a vector space…

A family $\mathcal{F}$ of spanning trees of the complete graph on $n$ vertices $K_n$ is \emph{$t$-intersecting} if any two members have a forest on $t$ edges in common. We prove an Erd\H{o}s--Ko--Rado result for $t$-intersecting families of…

A family $\mathcal{F}$ of $k$-subsets of an $n$-set is called $s$-almost $t$-intersecting if each member is $t$-disjoint with at most $s$ members. In this paper, we prove that, if $\left|\mathcal{F}\right|$ is maximum, then $\mathcal{F}$…

Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are said to be cross-$t$-intersecting if each set in $\mathcal{A}$ intersects each set in $\mathcal{B}$ in at least $t$ elements. An active problem in extremal set theory is to determine…

A subset $S$ of the alternating group on $n$ points is {\it intersecting} if for any pair of permutations $\pi,\sigma$ in $S$, there is an element $i\in \{1,\dots,n\}$ such that $\pi(i)=\sigma(i)$. We prove that if $S$ is intersecting, then…

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…

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 of sets is intersecting if every pair of its members has an element in common. Such a family of sets is called a star if some element is in every set of the family. Given a graph $G$, let $\mu(G)$ denote the size of the smallest…

Our main result is a new upper bound for the size of k-uniform, L-intersecting families of sets, where L contains only positive integers. We characterize extremal families in this setting. Our proof is based on the Ray-Chaudhuri--Wilson…

Given a point set $S$ in $\mathbb{R}^d$, a family of sets is $S$-intersecting if its members have a point in common in $S$. Recently, Edwards and Sober\'{o}n proved a fractional version of Halman's theorem for axis-parallel boxes, showing…

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 $X$…

Let $t$, $r$, $k$ and $n$ be positive integers and $\mathcal{F}$ a family of $k$-subsets of an $n$-set $V$. The family $ \CF $ is $ r $-wise $ t $-intersecting if for any $ F_1, \ldots, F_r \in \CF $, we have $ \abs{\cap_{i = 1}^{r}F_i}\gs…

We consider the following generalization of the seminal Erd\H{o}s-Ko-Rado theorem, due to Frankl. For some k>=2, let F be a k-wise intersecting family of r-subsets of an n element set X, i.e. for any k sets F1,...,Fk in F, their…

The Bollob\'as set pairs inequality is a fundamental result in extremal set theory with many applications. In this paper, for $n \geq k \geq t \geq 2$, we consider a collection of $k$ families $\mathcal{A}_i: 1 \leq i \leq k$ where…