The maximum sum of sizes of non-empty cross $L$-intersecting families

Abstract

Let $n$, $r$, and $k$ be positive integers such that $k,r\ge 2$, $L$ a non-empty subset of $[k]$, and ${\mathcal{F}}_i\subseteq \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ for $1\le i\le r$. We say that non-empty families ${\mathcal{F}}_1,{\mathcal{F}}_2,\dots ,{\mathcal{F}}_r$ are $r$-cross $L$-intersecting if $\left|{\bigcap }_{i=1}^r{F}_i\right|\in L$ for every choice of $F_i\in {\mathcal{F}}_i$ with $1\le i\le r$. They are called pairwise cross $L$-intersecting if $|A\cap B|\in L$ for all $A\in {\mathcal{F}}_i$, $B\in {\mathcal{F}}_j$ with $i\ne j$. If $r=2$, we simply say cross $L$-intersecting instead of $2$-cross $L$-intersecting or pairwise cross $L$-intersecting. In this paper, we determine the maximum possible sum of sizes of non-empty cross $L$-intersecting families ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ for all admissible $n$, $k$, and $L$, and we characterize all the extremal structures. We also establish the maximum value of the sum of sizes of families ${\mathcal{F}}_1,\dots ,{\mathcal{F}}_r$ that are both pairwise cross $L$-intersecting and $r$-cross $L$-intersecting, provided $n$ is sufficiently large and $L$ satisfies certain conditions. Furthermore, we characterize all such families attaining the maximum total size.