On extremal cross $t$-intersecting families with $t$-covering number conditions

Abstract

Let $n$, $k$ and $t$ be positive integers, and let $\mathcal{F}$ be a collection of $k$-subsets of $[n]=\{1,2,\dots ,n\}$. The $t$-covering number ${\tau }_t(\mathcal{F})$ of $\mathcal{F}$ is defined as the minimum size of a set $T$ such that $|F\cap T|\ge t$ for all $F\in \mathcal{F}$. For positive integers $k_1$ and $k_2$, let ${\mathcal{F}}_i$ be a collection of $k_i$-subsets of $[n]$ for $i\in \{1,2\}$. The families ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ are said to be cross $t$-intersecting if $|F_1\cap F_2|\ge t$ for all $F_1\in {\mathcal{F}}_1$ and $F_2\in {\mathcal{F}}_2$. When ${\mathcal{F}}_1={\mathcal{F}}_2$, ${\mathcal{F}}_1$ is called a $t$-intersecting family. In this paper, we first characterize the extremal structures of cross $t$-intersecting families ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ that maximize $|{\mathcal{F}}_1||{\mathcal{F}}_2|$ under the condition that ${\tau }_t({\mathcal{F}}_1)\ge t+1$ and ${\tau }_t({\mathcal{F}}_2)\ge t+1$. We then describe the maximal $t$-intersecting families with $t$-covering number $t+1$.