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

TL;DR

This paper characterizes the largest $t$-intersecting families of finite sets with a $t$-covering number at least $t+2$, extending previous results by Frankl for large $n$.

Contribution

It provides a characterization of extremal $t$-intersecting families with high $t$-covering number, generalizing earlier work by Frankl.

Findings

Identifies maximum size families under the given conditions.

Generalizes Frankl's results to broader parameters.

Provides structural insights into $t$-intersecting families.

Abstract

Let $\mathcal{F}\subseteq{[n]\choose k}$ 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 $|S\cap F|\geqslant t$ 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 $\tau_t(\mathcal{F})\geqslant t+2$ and $n$ is sufficiently large, thereby generalizing two results by Frankl.