Tuan Tran, Minimum saturated families of sets

A family $\mathcal F$ of subsets of [n] is called s-saturated if it contains no s pairwise disjoint sets, and moreover, no set can be added to $\mathcal F$ while preserving this property. More than 40 years ago, Erdős and Kleitman conjectured that an s-saturated family of subsets of [n] has size at least $(1 – 2^{-(s-1)})2^n$. It is a simple exercise to show that every s-saturated family has size at least $2^{n-1}$, 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. We prove such a result, showing that every s-saturated family of subsets of [n] has size at least $(1 – 1/s)2^n$. In this talk,  I will present two short proofs. This is joint work with M. Bucic, S. Letzter and B. Sudakov.