Minimum saturated families of sets

TL;DR

This paper proves a new lower bound on the size of s-saturated families of sets, showing they must contain at least (1 - 1/s) of all subsets, advancing the understanding of extremal set family properties.

Contribution

It establishes a tight lower bound for s-saturated families, improving previous bounds and introducing a multipartite version of the problem.

Findings

Every s-saturated family has size at least (1 - 1/s)2^n.

Multipartite version yields sum of sizes at least (s-1)2^n.

Bound is tight, exemplified by specific family constructions.

Abstract

We call a family $\mathcal{F}$ of subsets of $[n]$ $s$-saturated if it contains no $s$ pairwise disjoint sets, and moreover no set can be added to $\mathcal{F}$ while preserving this property (here $[n] = \{1,\ldots,n\}$). More than 40 years ago, Erd\H{o}s and Kleitman conjectured that an $s$-saturated family of subsets of $[n]$ has size at least $(1 - 2^{-(s-1)})2^n$. It is easy to show that every $s$-saturated family has size at least $\frac{1}{2}\cdot 2^n$, but, as was mentioned by Frankl and Tokushige, even obtaining a slightly better bound of $(1/2 + \varepsilon)2^n$, for some fixed $\varepsilon > 0$, seems difficult. In this note, we prove such a result, showing that every $s$-saturated family of subsets of $[n]$ has size at least $(1 - 1/s)2^n$. This lower bound is a consequence of a multipartite version of the problem, in which we seek a lower bound on $|\mathcal{F}_1| + \ldots + |\mathcal{F}_s|$ where $\mathcal{F}_1, \ldots, \mathcal{F}_s$ are families of subsets of $[n]$, such that there are no $s$ pairwise disjoint sets, one from each family $\mathcal{F}_i$, and furthermore no set can be added to any of the families while preserving this property. We show that $|\mathcal{F}_1| + \ldots + |\mathcal{F}_s| \ge (s-1)\cdot 2^n$, which is tight e.g.\ by taking $\mathcal{F}_1$ to be empty, and letting the remaining families be the families of all subsets of $[n]$.