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, Erd\H{o}s 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 21⋅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∣+…+∣Fs∣
where F1,…,Fs are families of subsets of [n],
such that there are no s pairwise disjoint sets, one from each family
Fi, and furthermore no set can be added to any of the families
while preserving this property. We show that ∣F1∣+…+∣Fs∣≥(s−1)⋅2n, which is tight e.g.\ by taking
F1 to be empty, and letting the remaining families be the families
of all subsets of [n].Comment: 8 page