Related papers: Uniform Set Systems with Uniform Witnesses

Since their introduction by Erd\H{o}s in 1950, covering systems (that is, finite collections of arithmetic progressions that cover the integers) have been extensively studied, and numerous questions and conjectures have been posed regarding…

Let $\lambda$ be the second largest eigenvalue in absolute value of a uniform random $d$-regular graph on $n$ vertices. It was famously conjectured by Alon and proved by Friedman that if $d$ is fixed independent of $n$, then…

We show that, for $pn \to \infty$, the largest set in a $p$-random sub-family of the power set of $\{1, \ldots, n\}$ containing no $k$-chain has size $( k - 1 + o(1) ) p \binom{n}{n/2}$ with high probability. This confirms a conjecture of…

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…

We consider a problem of maximizing the product of the sizes of two uniform cross-$t$-intersecting families of sets. We show that the value of this maximum is at most polynomially larger (in the size of a ground set) than a quantity…

The study of intersection problems on families of sets is one of the most important topics in extremal combinatorics. As we all know, the extremal problems involving certain intersection constraints are equivalent to that with the union…

A $(k,m)$-Furstenberg set is a subset $S \subset \mathbb{F}_q^n$ with the property that each $k$-dimensional subspace of $\mathbb{F}_q^n$ can be translated so that it intersects $S$ in at least $m$ points. Ellenberg and Erman proved that…

Erd\H{o}s-Ko-Rado (EKR) type theorems yield upper bounds on the sizes of families of sets, subject to various intersection requirements on the sets in the family. Stability versions of such theorems assert that if the size of a family is…

For integers $n\ge s\ge 2$ let $e(n,s)$ denote the maximum of $|\mathcal F|,$ where $\mathcal F$ is a family of subsets of an $n$-element set and $\mathcal F$ contains no $s$ pairwise disjoint members. Half a century ago, solving a…

Two families $\mathcal{F}$ and $\mathcal{G}$ are called cross-intersecting if for every $F\in \mathcal{F}$ and $G\in \mathcal{G}$, the intersection $F\cap G$ is non-empty. It is significant to determine the maximum sum of sizes of…

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…

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…

Paul Erd\H{o}s and L\'{a}szl\'{o} Lov\'{a}sz established by means of an example that there exists a maximal intersecting family of $k-$sets with approximately $(e-1)k!$ blocks. L\'{a}szl\'{o} Lov\'{a}sz conjectured that their example is…

Let $\mathbb{Z}^{+}$ be the set of positive integers. Let $C_{k}$ denote all subsets of $\mathbb{Z}^{+}$ such that neither of them contains $k + 1$ pairwise coprime integers and $C_k(n)=C_k\cap \{1,2,\ldots,n\}$. Let $f(n, k) =…

Consider a family $\mathcal{F}$ of $k$-subsets of an ambient $(k^2-k+1)$-set such that no pair of $k$-subsets in $\mathcal{F}$ intersects in exactly one element. In this short note we show that the maximal size of such $\mathcal{F}$ is…

Let $\mathcal{D}$ be a family of digraphs. A digraph $D$ is \emph{$\mathcal{D}$-saturated} if it contains no member of $\mathcal{D}$ as a subdigraph, but for any arc $e$ in the complement of $D$, the digraph $D + e$ contains some member of…

We investigate the asymptotic version of the Erd\H{o}s-Ko-Rado theorem for the random $k$-uniform hypergraph $\mathcal{H}^k(n,p)$. For $2 \leq k(n) \leq n/2$, let $N=\binom{n}k$ and $D=\binom{n-k}k$. We show that with probability tending to…

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…

The vertex set of the Kneser graph $K(n,k)$ is $V = \binom{[n]}{k}$ and two vertices are adjacent if the corresponding sets are disjoint. For any graph $F$, the largest size of a vertex set $U \subseteq V$ such that $K(n,k)[U]$ is $F$-free,…

Erd\H{o}s showed that every set of $n$ positive integers contains a subset of size at least $n/(k+1)$ containing no solutions to $x_1 + \cdots + x_k = y$. We prove that the constant $1/(k+1)$ here is best possible by showing that if $(F_m)$…