A probabilistic variant of Sperner's theorem and of maximal $r$-cover free families

Abstract

A family of sets is called $r$-\emph{cover free} if no set in the family is contained in the union of $r$ (or less) other sets in the family. A $1$-cover free family is simply an antichain with respect to set inclusion. Thus, Sperner's classical result determines the maximal cardinality of a $1$-cover free family of subsets of an $n$-element set. Estimating the maximal cardinality of an $r$-cover free family of subsets of an $n$-element set for $r>1$ was also studied. In this note we are interested in the following probabilistic variant of this problem. Let $S_0,S_1,\dots ,S_r$ be independent and identically distributed random subsets of an $n$-element set. Which distribution minimizes the probability that $S_0\subseteq {\bigcup }_{i=1}^r{S}_i$? A natural candidate is the uniform distribution on an $r$-cover-free family of maximal cardinality. We show that for $r=1$ such distribution is indeed best possible. In a complete contrast, we also show that this is far from being true for every $r>1$ and $n$ large enough.