On the union of intersecting families

TL;DR

This paper advances the understanding of the maximum size of unions of intersecting families of sets, providing tight bounds and characterizing extremal structures for large sets, improving previous results by Frankl and F"uredi.

Contribution

It proves a new upper bound on the union size of intersecting families for fixed r and k, with a precise characterization of extremal families, using influence-based isoperimetric methods.

Findings

Established an upper bound for the union of r intersecting families.

Characterized the structure of extremal families achieving equality.

Improved previous bounds by Frankl and F"uredi for larger k.

Abstract

A family of sets is said to be \emph{intersecting} if any two sets in the family have nonempty intersection. In 1973, Erd\H{o}s raised the problem of determining the maximum possible size of a union of $r$ different intersecting families of $k$-element subsets of an $n$-element set, for each triple of integers $(n,k,r)$. We make progress on this problem, proving that for any fixed integer $r \geq 2$ and for any $k \leq (\tfrac{1}{2}-o(1))n$, if $X$ is an $n$-element set, and $\mathcal{F} = \mathcal{F}_1 \cup \mathcal{F}_2 \cup \ldots \cup \mathcal{F}_r$, where each $\mathcal{F}_i$ is an intersecting family of $k$-element subsets of $X$, then $|\mathcal{F}| \leq {n \choose k} - {n-r \choose k}$, with equality only if $\mathcal{F} = \{S \subset X:\ |S|=k,\ S \cap R \neq \emptyset\}$ for some $R \subset X$ with $|R|=r$. This is best possible up to the size of the $o(1)$ term, and improves a 1987 result of Frankl and F\"uredi, who obtained the same conclusion under the stronger hypothesis $k < (3-\sqrt{5})n/2$, in the case $r=2$. Our proof utilises an isoperimetric, influence-based method recently developed by Keller and the authors.