THE ANTIPODAL LAYERS PROBLEM

For n > 2k and [n] = {1,2,...,n), let the bipartite graph M(n,k) have vertices {A subset-of [n] parallel-to A\ = k or n - k} and edges {(A, B) \ A subset-of B). It has been conjectured that M2k+1,k (the middle two levels of the Boolean lattice l2k+1) is Hamiltonian, and we conjecture the same for arbitrary n. Here we show that the conjecture holds for n bigger than roughly k2, with k large enough. We also define a new product between ranked posets, giving rise to many new representations of M2k+1,k.