The number of maximal independent sets in the Hamming cube

TL;DR

This paper determines the asymptotic number of maximal independent sets in the n-dimensional Hamming cube, confirming a conjecture and connecting combinatorial structures with isoperimetric properties.

Contribution

It proves the asymptotic count of maximal independent sets in the Hamming cube, establishing both lower and upper bounds and confirming a conjecture.

Findings

Number of maximal independent sets asymptotically equals 2n * 2^{N/4}.

Established connections between independent sets and induced matchings.

Utilized stability and isoperimetric results in the proof.

Abstract

Let $Q_n$ be the $n$-dimensional Hamming cube and $N=2^n$. We prove that the number of maximal independent sets in $Q_n$ is asymptotically \[2n2^{N/4},\] as was conjectured by Ilinca and the first author in connection with a question of Duffus, Frankl and R\"odl. The value is a natural lower bound derived from a connection between maximal independent sets and induced matchings. The proof that it is also an upper bound draws on various tools, among them "stability" results for maximal independent set counts and old and new results on isoperimetric behavior in $Q_n$.