Related papers: On sizes of 1-cross intersecting set pair systems

Frankl--Pach and Erd\H{o}s conjectured that any $(d+1)$-uniform set family $\mathcal{F}\subseteq \binom{[n]}{d+1}$ with VC-dimension at most $d$ has size at most $\binom{n-1}{d}$ when $n$ is sufficiently large. Ahlswede and Khachatrian…

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…

Two families $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting if $A\cap B\ne \emptyset$ for any $A\in \mathcal{A}$ and $B\in \mathcal{B}$. We call $t$ families $\mathcal{A}_1, \mathcal{A}_2,\dots, \mathcal{A}_t$ pairwise…

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 show that any set of $n$ points in general position in the plane determines $n^{1-o(1)}$ pairwise crossing segments. The best previously known lower bound, $\Omega\left(\sqrt n\right)$, was proved more than 25 years ago by Aronov, Erd\H…

A family $\mathcal{A}$ of sets is said to be intersecting if every two sets in $\mathcal{A}$ intersect. Two families $\mathcal{A}$ and $\mathcal{B}$ are said to be cross-intersecting if each set in $\mathcal{A}$ intersects each set in…

In this paper, we derive a tight upper bound for the size of an intersecting $k$-Sperner family of subspaces of the $n$-dimensional vector space $\mathbb{F}_{q}^{n}$ over finite field $\mathbb{F}_{q}$ which gives a $q$-analogue of the…

Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be two families of subsets of an $n$-element set. We say that $\mathcal{F}_1$ and $\mathcal{F}_2$ are multiset-union-free if for any $A,B\in \mathcal{F}_1$ and $C,D\in \mathcal{F}_2$ the multisets…

A family $\mathcal{F}\subset \binom{[n]}{k}$ is called an intersecting family if $F\cap F'\neq \emptyset$ for all $F,F'\in \mathcal{F}$. If $\cap \mathcal{F}\neq \emptyset$ then $\mathcal{F}$ is called a star. The diversity of an…

Let $q=p^\alpha$ be a fixed prime power, $k\geq 2$ be an integer. We give a new upper bound for the size of $k$-wise $q$-modular $L$-avoiding $L$-intersecting set systems, where $L$ is any proper subset of $\{0, \ldots , q-1\}$. Our proof…

Fermionic superfluidity requires the formation of pairs. The actual size of these fermion pairs varies by orders of magnitude from the femtometer scale in neutron stars and nuclei to the micrometer range in conventional superconductors.…

Let F be a set system on [n] with all sets having k elements and every pair of sets intersecting. The celebrated theorem of Erdos-Ko-Rado from 1961 says that any such system has size at most ${n-1 \choose k-1}$. A natural question, which…

Let $n > k > 1$ be integers, $[n] = \{1, \ldots, n\}$. Let $\mathcal F$ be a family of $k$-subsets of~$[n]$. The family $\mathcal F$ is called intersecting if $F \cap F' \neq \emptyset$ for all $F, F' \in \mathcal F$. It is called almost…

In this paper, we analyze the structure of maximal sets of $k$-dimensional spaces in $\mathrm{PG}(n,q)$ pairwise intersecting in at least a $(k-2)$-dimensional space, for $3 \leq k\leq n-2$. We give an overview of the largest examples of…

Denote by $f_D(n)$ the maximum size of a set family $\mathcal{F}$ on $[n] \stackrel{\mbox{\normalfont\tiny def}}{=} \{1, \dots, n\}$ with distance set $D$. That is, $|A \bigtriangleup B| \in D$ holds for every pair of distinct sets $A, B…

We study the problem of determining the size of the largest intersecting $P$-free family for a given partially ordered set (poset) $P$. In particular, we find the exact size of the largest intersecting $B$-free family where $B$ is the…

A family $\mathcal{F}$ on ground set $[n]:=\{1,2,\ldots, n\}$ is maximal $k$-wise intersecting if every collection of at most $k$ sets in $\mathcal{F}$ has non-empty intersection, and no other set can be added to $\mathcal{F}$ while…

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…

A $\left(n,\ell,\gamma\right)$-sharing set family of size $m$ is a family of sets $S_1,\ldots,S_m\subseteq [n]$ s.t. each set has size $\ell$ and each pair of sets shares at most $\gamma$ elements. We let $m\left(n,\ell,\gamma\right)$…

A family $\mathcal{F}$ on ground set $\{1,2,\ldots, n\}$ is maximal $k$-wise intersecting if every collection of $k$ sets in $\mathcal{F}$ has non-empty intersection, and no other set can be added to $\mathcal{F}$ while maintaining this…