Minimum saturated families of sets

We call a family F of subsets of [n] s-saturated if it contains no s pairwise disjoint sets, and moreover no set can be added to F while preserving this property (here [n] = {1, . . . , n}). More than 40 years ago, Erdős and Kleitman conjectured that an s -saturated family of subsets of [n] has size at least (1 − 2−^(s−1))2ⁿ. It is easy to show that every s -saturated family has size at least 1/2.2ⁿ, but, as was mentioned by Frankl and Tokushige, even obtaining a slightly better bound of (1/2 + ε)2ⁿ, for some fixed ε > 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)2ⁿ. This lower bound is a consequence of a multipartite version of the problem, in which we seek a lower bound on |F₁| + · · · + |F_s| where F₁, . . . ,F_s are families of subsets of [n], such that there are no s pairwise disjoint sets, one from each family F_i, and furthermore no set can be added to any of the families while preserving this property. We show that |F₁| + · · · + |F_s| ≥ (s − 1) · 2ⁿ, which is tight, for example, by taking F₁ to be empty, and letting the remaining families be the families of all subsets of [n].