On Dedekind's problem, a sparse version of Sperner's theorem, and antichains of a given size in the Boolean lattice

TL;DR

This paper applies a statistical physics-inspired cluster expansion method to analyze Dedekind's problem, providing new estimates for the number and structure of antichains in Boolean lattices, and establishing a sparse Sperner's theorem variant.

Contribution

It introduces a novel approach to Dedekind's problem using cluster expansion, yielding detailed estimates and a sparse Sperner's theorem with sharp thresholds.

Findings

Derived new estimates for the total number of antichains in $B_n$.

Identified the typical structure of antichains in Boolean lattices.

Established a sharp threshold for the property that most antichains are in the middle layer.

Abstract

Dedekind's problem, dating back to 1897, asks for the total number $\psi(n)$ of antichains contained in the Boolean lattice $B_n$ on $n$ elements. We study Dedekind's problem using a recently developed method based on the cluster expansion from statistical physics and as a result, obtain several new results on the number and typical structure of antichains in $B_n$. We obtain detailed estimates for both $\psi(n)$ and the number of antichains of size $\beta \binom{n}{\lfloor n/2 \rfloor}$ for any fixed $\beta>0$. We also establish a sparse version of Sperner's theorem: we determine the sharp threshold and scaling window for the property that almost every antichain of size $m$ is contained in a middle layer of $B_n$.