A note on saturation for $k$-wise intersecting families

TL;DR

This paper investigates the size of maximal $k$-wise intersecting families of subsets, establishing an upper bound that aligns with known lower bounds and resolving a longstanding question in combinatorics.

Contribution

It proves an upper bound on the size of maximal $k$-wise intersecting families, matching known lower bounds up to a constant factor, and addresses an open problem posed by Erd"H{o}s and Kleitman.

Findings

Maximal $k$-wise intersecting families have size $O(2^{n/(k-1)})$.

The bound matches the best known lower bound up to a constant.

The work resolves an old open question in the theory of intersecting families.

Abstract

A family $\mathcal{F}$ of subsets of $\{1,\dots,n\}$ is called $k$-wise intersecting if any $k$ members of $\mathcal{F}$ have non-empty intersection, and it is called maximal $k$-wise intersecting if no family strictly containing $\mathcal{F}$ satisfies this condition. We show that for each $k\geq 2$ there is a maximal $k$-wise intersecting family of size $O(2^{n/(k-1)})$. Up to a constant factor, this matches the best known lower bound, and answers an old question of Erd\H{o}s and Kleitman, recently studied by Hendrey, Lund, Tompkins, and Tran.