Related papers: On hierarchically closed fractional intersecting f…

Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_q$, and ${V\brack k}$ denote the family of all $k$-dimensional subspaces of $V$. The families $\mathcal{F}_1\subseteq{V\brack k_1},\mathcal{F}_2\subseteq{V\brack…

A family of perfect matchings of $K_{2n}$ is $intersecting$ if any two of its members have an edge in common. It is known that if $\mathcal{F}$ is family of intersecting perfect matchings of $K_{2n}$, then $|\mathcal{F}| \leq (2n-3)!!$ and…

Given a sequence of positive integers $p = (p_1, . . ., p_n)$, let $S_p$ denote the family of all sequences of positive integers $x = (x_1,...,x_n)$ such that $x_i \le p_i$ for all $i$. Two families of sequences (or vectors), $A,B \subseteq…

For a graph property $\mathcal{P}$ and a common vertex set $V = \{1, 2, \ldots, n\}$, a family of graphs on $V$ is \emph{$\mathcal{P}$-intersecting} iff $G \cap H$ satisfies $\mathcal{P}$ for all $G,H$ in the family. Addressing a question…

A family $\mathcal C$ of sets is hereditary if whenever $A\in \mathcal C$ and $B\subset A$, we have $B\in \mathcal C$. Chv\'atal conjectured that the largest intersecting subfamily of a hereditary family is the family of all sets containing…

It is well known that an intersecting family of subsets of an n-element set can contain at most 2^(n-1) sets. It is natural to wonder how `close' to intersecting a family of size greater than 2^(n-1) can be. Katona, Katona and Katona…

Let $V$ be an $(n+\ell)$-dimensional vector space over a finite field, and $W$ a fixed $\ell$-dimensional subspace of $V$. Write ${V\brack n,0}$ to be the set of all $n$-dimensional subspaces $U$ of $V$ satisfying $\dim(U\cap W)=0$. 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…

The Frankl conjecture, also known as the union-closed sets conjecture, states that in any finite non-empty union-closed family, there exists an element in at least half of the sets. From an optimization point of view, one could instead…

Let $k\ge d\ge 3$ be fixed. Let $\mathcal{F}$ be a $k$-uniform family on $[n]$. Then $\mathcal{F}$ is $(d,s)$-conditionally intersecting if it does not contain $d$ sets with union of size at most $s$ and empty intersection. Answering a…

The families $\mathcal{A}$ and $\mathcal{B}$ are cross intersecting if $A\cap B\ne \emptyset$ for any $A\in \mathcal{A}$ and $B\in \mathcal{B}$. Let $t\geq 2$ and $k_1\geq k_2\geq \cdots \geq k_t$. We say that $(\mathcal{F}_1, \dots,…

A set family ${\cal F}$ is called intersecting if every two members of ${\cal F}$ intersect, and it is called uniform if all members of ${\cal F}$ share a common size. A uniform family ${\cal F} \subseteq \binom{[n]}{k}$ of $k$-subsets of…

Let $m\geq 2$, $n$ be positive integers, and $R_i=\{k_{i,1} >k_{i,2} >\cdots> k_{i,t_i}\}$ be subsets of $[n]$ for $i=1,2,\ldots,m$. The families $\mathcal{F}_1\subseteq \binom{[n]}{R_1},\mathcal{F}_2\subseteq…

Two families $\mathcal A\subseteq\binom{[n]}{k}$ and $\mathcal B\subseteq\binom{[n]}{\ell}$ are called cross-$t$-intersecting if $|A\cap B|\geq t$ for all $A\in\mathcal A$, $B\in\mathcal B$. Let $n$, $k$ and $\ell$ be positive integers such…

Let $ n\geqslant t\geqslant 1$ and $ \mathcal{A}_1, \mathcal{A}_2, \ldots, \mathcal{A}_m \subseteq 2^{[n]}$ be non-empty families. We say that they are pairwise cross $t$-intersecting if $|A_i\cap A_j|\geqslant t$ holds for any $A_i\in…

Let $V$ be a finite dimensional vector space over a finite field. Suppose that $\mathscr{F}_1$, $\mathscr{F}_2$, $\dots$, $\mathscr{F}_r$ are $r$-cross $t$-intersecting families of $k$-subspaces of $V$. In this paper, we determine the…

Given a family of sets on the plane, we say that the family is intersecting if for any two sets from the family their interiors intersect. In this paper, we study intersecting families of triangles with vertices in a given set of points. In…

Let $\mathcal{F}$ be a family of $n$ axis-parallel boxes in $\mathbb{R}^d$ and $\alpha\in (1-1/d,1]$ a real number. There exists a real number $\beta(\alpha )>0$ such that if there are $\alpha {n\choose 2}$ intersecting pairs in…

Let $\mathbb N_0$ be the set of non-negative integers, and let $P(n,l)$ denote the set of all weak compositions of $n$ with $l$ parts, i.e., $P(n,l)=\{ (x_1,x_2,\dots, x_l)\in\mathbb N_0^l\ :\ x_1+x_2+\cdots+x_l=n\}$. For any element…

Families $\mathcal{A}_1, \mathcal{A}_2, ..., \mathcal{A}_k$ of sets are said to be \emph{cross-intersecting} if for any $i$ and $j$ in $\{1, 2, ..., k\}$ with $i \neq j$, any set in $\mathcal{A}_i$ intersects any set in $\mathcal{A}_j$. For…