On the number of antichains in $\{0,1,2\}^n$

TL;DR

This paper derives precise asymptotic formulas for counting antichains in the poset 0,1,2^n, advancing understanding of combinatorial structures with complex partial orderings.

Contribution

It provides the first exact asymptotics for the number of antichains in 0,1,2^n, using novel graph-container and isoperimetric techniques.

Findings

Established precise asymptotics for antichains in 0,1,2^n

Developed a graph-container lemma for irregular graphs

Applied isoperimetric inequalities and statistical physics methods

Abstract

We provide precise asymptotics for the number of antichains in the poset $\{0,1,2\}^n$, answering a question of Sapozhenko. Finding improved estimates for this number was also a problem suggested by Noel, Scott, and Sudakov, who obtained asymptotics for the logarithm of the number. Key ingredients for the proof include a graph-container lemma to bound the number of expanding sets in a class of irregular graphs, isoperimetric inequalities for generalizations of the Boolean lattice, and methods from statistical physics based on the cluster expansion.