Entropy, independent sets and antichains: A new approach to Dedekind’s problem

For n-regular, N-vertex bipartite graphs with bipartition A∪ B, a precise bound is given for the sum over independent sets I of the quantity μ |I∩ A|λ |I∩ B|. (In other language, this is bounding the partition function for certain instances of the hard-core model.) This result is then extended to graded partially ordered sets, which in particular provides a simple proof of a well-known bound for Dedekind’s Problem given by Kleitman and Markowsky in 1975.