Some intersection theorems for finite sets

Abstract

Let $n$, $r$, $k_1,\dots ,k_r$ and $t$ be positive integers with $r\ge 2$, and ${\mathcal{F}}_i(1\le i\le r)$ a family of $k_i$-subsets of an $n$-set $V$. The families ${\mathcal{F}}_1,{\mathcal{F}}_2,\dots ,{\mathcal{F}}_r$ are said to be $r$-cross $t$-intersecting if $|F_1\cap F_2\cap \cdots \cap F_r|\ge t$ for all $F_i\in {\mathcal{F}}_i(1\le i\le r),$ and said to be non-trivial if $|{\cap }_{1\le i\le r}{\cap }_{F\in {\mathcal{F}}_i}F|<t$. If the $r$-cross $t$-intersecting families ${\mathcal{F}}_1,\dots ,{\mathcal{F}}_r$ satisfy ${\mathcal{F}}_1=\cdots ={\mathcal{F}}_r=\mathcal{F}$, then $\mathcal{F}$ is well known as $r$-wise $t$-intersecting family. In this paper, we describe the structure of non-trivial $r$-wise $t$-intersecting families with maximum size, and give a stability result for these families. We also determine the structure of non-trivial $2$-cross $t$-intersecting families with maximum product of their sizes.

28 pages. arXiv admin note: text overlap with arXiv:2201.06339