Counting maximal antichains and independent sets

TL;DR

This paper provides asymptotic estimates for the number of maximal antichains in Boolean algebras and maximal independent sets in hypercube graphs, advancing understanding of combinatorial structures.

Contribution

It offers new asymptotic formulas for counting these structures, extending previous questions and including bounds for regular and biregular graphs.

Findings

Asymptotics for maximal antichains in Boolean algebras

Asymptotics for maximal independent sets in hypercube graphs

Upper bounds for independent sets in regular graphs

Abstract

Answering several questions of Duffus, Frankl and R\"odl, we give asymptotics for the logarithms of (i) the number of maximal antichains in the n-dimensional Boolean algebra and (ii) the numbers of maximal independent sets in the covering graph of the n-dimensional hypercube and certain natural subgraphs thereof. The results in (ii) are implied by more general upper bounds on the numbers of maximal independent sets in regular and biregular graphs. We also mention some stronger possibilities involving actual rather than logarithmic asymptotics.