Extremal $t$-intersecting families for finite sets with $t$-covering number at least $t+2$

Abstract

Let $\mathcal{F}\subseteq \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ be a $t$-intersecting family. Define the $t$-covering number ${\tau }_t(\mathcal{F})$ of $\mathcal{F}$ as the minimum size of a subset $S$ of $[n]$ with $|S\cap F|⩾t$ for each $F\in \mathcal{F}$. In this paper, we characterize $\mathcal{F}$ for which $|\mathcal{F}|$ takes the maximum value under the condition that ${\tau }_t(\mathcal{F})⩾t+2$ and $n$ is sufficiently large, thereby generalizing two results by Frankl.