Note on the number of balanced independent sets in the Hamming cube (Q2144319)

Summary: 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''.