Independent sets in the middle two layers of Boolean lattice

For an odd integer n=2d?1, let B(n,d) be the subgraph of hypercube Qn induced by two largest layers. In this paper, we describe typical structure independent sets in and give precise asymptotics on number them. The proofs use Sapozhenko's graph container method a recently developed Jenssen Perkins, which combines lemma with cluster expansion for polymer models from statistical physics.