Maximal $3$-wise intersecting families

TL;DR

This paper determines the minimal size of a maximal 3-wise intersecting family on a large set, showing it is uniquely achieved by a specific partition-based construction, thus solving a longstanding combinatorial problem.

Contribution

It provides the first exact characterization of the minimal size and structure of maximal 3-wise intersecting families for large n, answering a question posed in 1974.

Findings

The minimal family is formed by all supersets of a bipart partition of the ground set.

This minimal family is unique for sufficiently large n.

The result extends understanding of intersection properties in combinatorial families.

Abstract

A family $\mathcal{F}$ on ground set $[n]:=\{1,2,\ldots, n\}$ is maximal $k$-wise intersecting if every collection of at most $k$ sets in $\mathcal{F}$ has non-empty intersection, and no other set can be added to $\mathcal{F}$ while maintaining this property. In 1974, Erd\H{o}s and Kleitman asked for the minimum size of a maximal $k$-wise intersecting family. We answer their question for $k=3$ and sufficiently large $n$. We show that the unique minimum family is obtained by partitioning the ground set $[n]$ into two sets $A$ and $B$ with almost equal sizes and taking the family consisting of all the proper supersets of $A$ and of $B$.