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