Note on the number of balanced independent sets in the Hamming cube

TL;DR

This paper investigates the count of balanced independent sets in the d-dimensional Hamming cube, revealing that their logarithm scales as \\(1-\Theta(1/\sqrt d)\\) times half the total vertices, using an enhanced graph container lemma.

Contribution

It introduces an improved version of Sapozhenko's graph container lemma to analyze balanced independent sets in the Hamming cube.

Findings

Logarithm of balanced independent sets scales as \\(1-\Theta(1/\sqrt d)\\) times N/2.

Provides asymptotic estimate for the number of balanced independent sets.

Enhances combinatorial techniques with a refined graph container lemma.

Abstract

Let $Q_d$ be the $d$-dimensional Hamming cube and $N=|V(Q_d)|=2^d$. An independent set $I$ in $Q_d$ is called balanced if $I$ contains the same number of even and odd vertices. We show that the logarithm of the number of balanced independent sets in $Q_d$ is \[(1-\Theta(1/\sqrt d))N/2.\] The key ingredient of the proof is an improved version of "Sapozhenko's graph container lemma."