Related papers: On hierarchically closed fractional intersecting f…

A family of subsets of $\{1,\ldots,n\}$ is called {\it intersecting} if any two of its sets intersect. A classical result in extremal combinatorics due to Erd\H{o}s, Ko, and Rado determines the maximum size of an intersecting family of…

For a family of sets $\mathcal{F}$, let $\omega(\mathcal{F}):=\sum_{\{A,B\}\subset \mathcal{F}}|A\cap B|$. In this paper, we prove that provided $n$ is sufficiently large, for any $\mathcal{F}\subset \binom{[n]}{k}$ with $|\mathcal{F}|=m$,…

An intersecting $r$-uniform straight line system is an intersecting linear system whose lines consist of $r$ points on straight line segment of $\mathbb{R}^2$ and any two lines share a point. Recently, the author [A. V\'azquez-\'Avila,…

For a left-compressed intersecting family \A contained in [n]^(r) and a set X contained in [n], let \A(X) = {A in \A : A intersect X is non-empty}. Borg asked: for which X is |\A(X)| maximised by taking \A to be all r-sets containing the…

Let $\mathscr{P}$ be a symplectic polar space over a finite field $\mathbb{F}_q$, and $\mathscr{P}_m$ denote the set of all $m$-dimensional subspaces in $\mathscr{P}$. We say a $t$-intersecting subfamily of $\mathscr{P}_m$ is trivial if…

We prove that for $n$ sufficiently large, if $A$ is a family of permutations of $\{1,2,\ldots,n\}$ with no two permutations in $\mathcal{A}$ agreeing exactly once, then $|\mathcal{A}| \leq (n-2)!$, with equality holding only if…

This paper resolves two open problems in extremal set theory. For a family $\mathcal{F} \subseteq 2^{[n]}$ and $i, j\in [n]$, we denote $\mathcal{F} (i,\bar{j})=\{F\backslash\{i\}: F\in \mathcal{F}, F\cap\{i,j\}=\{i\}\}$. The sturdiness…

For any $\epsilon>0$ and $n>(1+\epsilon)t$, $n>n_0(\epsilon)$ we determine the size of the largest $t$-intersecting family of permutations, as well as give a sharp stability result. This resolves a conjecture of Ellis, Friedgut and Pilpel…

Let $ k, n \in \mathbb{N}^+ $ and $ m \in \mathbb{N}^+ \cup \{\infty \} $. A $ k $-multiset in $ [n]_m $ is a $ k $-set whose elements are integers from $ \{1, 2, \ldots, n\} $, and each element is allowed to have at most $ m $ repetitions.…

A family $\mathcal{F}$ of subsets of $[n]=\{1,2,\ldots,n\}$ shatters a set $A \subseteq [n]$ if for every $A' \subseteq A$ there is an $F \in \mathcal{F}$ such that $F \cap A=A'$. We develop a framework to analyze $f(n,k,d)$, the maximum…

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…

Our aim in this note is to show that, for any $\epsilon>0$, there exists a union-closed family $\mathcal F$ with (unique) smallest set $S$ such that no element of $S$ belongs to more than a fraction $\epsilon$ of the sets in $\mathcal F$.…

Let $n$ be a positive integer, let $0<p\leqslant p'\leqslant \frac{1}{2}$, and let $\ell \leqslant pn$ be a nonnegative integer. We prove that if $\mathcal{F},\mathcal{G}\subseteq \{0,1\}^n$ are two families whose cross intersections forbid…

A classic theorem in combinatorial design theory is Fisher's inequality, which states that a family $\mathcal F$ of subsets of $[n]$ with all pairwise intersections of size $\lambda$ can have at most $n$ non-empty sets. One may weaken the…

A linear system is a pair $(P,\mathcal{L})$ where $\mathcal{L}$ is a family of subsets on a ground finite set $P$, such that $|l\cap l^\prime|\leq 1$, for every $l,l^\prime \in \mathcal{L}$. The elements of $P$ and $\mathcal{L}$ are called…

Let F be a family of subsets of an n-element set not containing four distinct members such that A union B is contained in C intersect D. It is proved that the maximum size of F under this condition is equal to the sum of the two largest…

Ryser's Conjecture states that for any $r$-partite $r$-uniform hypergraph, the vertex cover number is at most $r{-}1$ times the matching number. This conjecture is only known to be true for $r\leq 3$ in general and for $r\leq 5$ if the…

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…

For integers $r$ and $n$, where $n$ is sufficiently large, and for every set $X \subseteq [n]$ we determine the maximal left-compressed intersecting families $\mathcal{A}\subseteq \binom{[n]}{r}$ which achieve maximum hitting with $X$ (i.e.…

Let $\omega(\mathcal{F})=\sum_{\{A,B\}\subset\mathcal{F}}|A\cap B|$ and $\omega(\mathcal{A},\mathcal{B})=\sum_{(A,B)\in \mathcal{A}\times \mathcal{B}}|A\cap B|$. A family $\mathcal{F}$ is intersecting if $F_1\cap F_2\neq \emptyset$ for any…