A generalization of Sperner's theorem and an application to graph orientations

A generalization of Sperner's theorem is established: For a Multifamily M = {Y(1),..., Y(p)} of subsets of {1,..., n) in which the repetition of subsets is allowed, a sharp lower bound for the number phi(M) of ordered pairs (i, j) satisfying i not equal j and Y(i) subset of Y(j) is determined. As an application, the minimum average distance of orientations of complete bipartite graphs is determined. (C) 2007 Elsevier B.V. All rights reserved