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

TL;DR

This work determines the maximum size of a $t$-intersecting family $\mathcal{F}\subseteq{[n]\choose k}$ under the nontrivial constraint $τ_t(\mathcal{F})\ge t+2$ for large $n$ and $k\ge t+3$. By introducing the auxiliary $t$-cover set system $\mathcal{T}_t(\mathcal{F})$ and performing a careful case analysis based on $τ_t(\mathcal{T}_t(\mathcal{F}))\in\{t,t+1,t+2\}$, the authors derive three explicit extremal constructions $\mathcal{F}_1,\mathcal{F}_2,\mathcal{F}_3$ and prove sharp upper bounds $|\mathcal{F}|\le\max\{f_1(n,k,t),f_2(n,k,t),f_3(n,k,t)\}$. Equality occurs exactly for families arising from these constructions, thereby extending classical Erdős–Ko–Rado-type results to nontrivial t-intersecting families with prescribed covering numbers. The approach combines structural analysis of $t$-covers, cross-intersection bounds, and Frankl’s recent results to achieve a complete extremal classification for large $n$. The findings deepen our understanding of how covering constraints shape extremal set systems and provide precise templates for maximal nontrivial $t$-intersecting families.

Abstract

Let $\mathcal{F}\subseteq{[n]\choose k}$ be a $t$-intersecting family. Define the $t$-covering number $τ_t(\mathcal{F})$ 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 $τ_t(\mathcal{F})\geqslant t+2$ and $n$ is sufficiently large, thereby generalizing two results by Frankl.

Theorems & Definitions (27)

• Theorem 1.1
• Lemma 2.1
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Proposition 2.4
• Theorem 2.5
• Lemma 2.6
• proof