Note on the Number of Balanced Independent Sets in the Hamming Cube

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".