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arXiv · 2205.11630

A sharp threshold for a random version of Sperner's Theorem

József Balogh·Robert A. Krueger
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Abstract

The Boolean lattice $\mathcal{P}(n)$ consists of all subsets of $[n] = \{1,\dots, n\}$ partially ordered under the containment relation. Sperner's Theorem states that the largest antichain of the Boolean lattice is given by a middle layer: the collection of all sets of size $\lfloor{n/2}\rfloor$, or also, if $n$ is odd, the collection of all sets of size $\lceil{n/2}\rceil$. Given $p$, choose each subset of $[n]$ with probability $p$ independently. We show that for every constant $p>3/4$, the largest antichain among these subsets is also given by a middle layer, with probability tending to $1$ as $n$ tends to infinity. This $3/4$ is best possible, and we also characterize the largest antichains for every constant $p>1/2$. Our proof is based on some new variations of Sapozhenko's graph container method.

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József Balogh, Robert A. Krueger. 2022-05-23. A sharp threshold for a random version of Sperner's Theorem. https://arxiv.org/abs/2205.11630

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