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

TL;DR

This paper explores a probabilistic variant of Sperner's theorem and maximal $r$-cover free families, analyzing how to minimize the probability that a random set is contained in the union of others, revealing contrasting results for different values of r.

Contribution

It introduces a probabilistic approach to $r$-cover free families, showing optimal distributions for $r=1$ and contrasting behaviors for $r>1$ with large $n$.

Findings

For $r=1$, the uniform distribution on a maximal $1$-cover free family minimizes the probability.

For $r>1$ and large $n$, the uniform distribution is not optimal for minimizing the containment probability.

The results highlight a fundamental difference between the cases $r=1$ and $r>1$ in probabilistic set 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 $S_0,S_1,\ldots, S_r$ 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.