Maximal 3-wise Intersecting Families with Minimum Size: the Odd Case

TL;DR

This paper determines the minimal size of a maximal 3-wise intersecting family on an odd-sized set, showing it is uniquely achieved by a specific partition-based family, extending previous even-sized set results.

Contribution

It provides the first characterization of minimal maximal 3-wise intersecting families for large odd sets, completing the understanding for all large n.

Findings

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

The minimal size is achieved by a family based on partitioning into two nearly equal parts.

The proof uses a stability result related to 2-generator set systems.

Abstract

A family $\mathcal{F}$ on ground set $\{1,2,\ldots, n\}$ is maximal $k$-wise intersecting if every collection of $k$ sets in $\mathcal{F}$ has non-empty intersection, and no other set can be added to $\mathcal{F}$ while maintaining this property. Erd\H{o}s and Kleitman asked for the minimum size of a maximal $k$-wise intersecting family. Complementing earlier work of Hendrey, Lund, Tompkins and Tran, who answered this question for $k=3$ and large even $n$, we answer it for $k=3$ and large odd $n$. We show that the unique minimum family is obtained by partitioning the ground set into two sets $A$ and $B$ with almost equal sizes and taking the family consisting of all the proper supersets of $A$ and of $B$. A key ingredient of our proof is the stability result by Ellis and Sudakov about the so-called $2$-generator set systems.