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arXiv · 1803.07591

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

Noga Alon·Shoni Gilboa·Shay Gueron
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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.

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Noga Alon, Shoni Gilboa, Shay Gueron. 2018-03-20. A probabilistic variant of Sperner's theorem and of maximal $r$-cover free families. https://doi.org/10.1016/j.disc.2020.112027

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