On an inverse problem of the Erd\H{o}s-Ko-Rado type theorems

TL;DR

This paper investigates the structure of set families that maximize total intersection, extending classical intersection theorems by characterizing extremal families and their stability using combinatorial methods.

Contribution

It provides new combinatorial characterizations of families maximizing total intersection, linking intersection properties with extremal structure in a quantitative setting.

Findings

Optimal families are t-intersecting for large enough n and proper sizes.

Upper bounds on total intersection for various family sizes.

Unique optimal structures identified for certain family sizes.

Abstract

A family of subsets $\mathcal{F}\subseteq {[n]\choose k}$ is called intersecting if any two of its members share a common element. Consider an intersecting family, a direct problem is to determine its maximal size and the inverse problem is to characterize its extremal structure and its corresponding stability. The famous Erd\H{o}s-Ko-Rado theorem answered both direct and inverse problems and led the era of studying intersection problems for finite sets. In this paper, we consider the following quantitative intersection problem which can be viewed an inverse problem for Erd\H{o}s-Ko-Rado type theorems: For $\mathcal{F}\subseteq {[n]\choose k}$, define its \emph{total intersection} as $\mathcal{I}(\mathcal{F})=\sum_{F_1,F_2\in \mathcal{F}}|F_1\cap F_2|$. Then, what is the structure of $\mathcal{F}$ when it has the maximal total intersection among all families in ${[n]\choose k}$ with the same family size? Using a pure combinatorial approach, we provide two structural characterizations of the optimal family of given size that maximizes the total intersection. As a consequence, for $n$ large enough and $\mathcal{F}$ of proper size, these characterizations show that the optimal family $\mathcal{F}$ is indeed $t$-intersecting ($t\geq 1$). To a certain extent, this reveals the relationship between properties of being intersecting and maximizing the total intersection. Also, we provide an upper bound on $\mathcal{I}(\mathcal{F})$ for several ranges of $|\mathcal{F}|$ and determine the unique optimal structure for families with sizes of certain values.