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

TL;DR

This paper establishes a sharp threshold at p=3/4 for the emergence of a largest antichain in a random subset of the Boolean lattice, extending Sperner's Theorem to probabilistic settings.

Contribution

It identifies the precise probability threshold where the largest antichain is typically a middle layer, using novel variations of the graph container method.

Findings

For p > 3/4, the largest antichain is a middle layer with high probability.

The threshold p=3/4 is proven to be optimal.

Characterization of largest antichains for all p > 1/2.

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.