The Saturation Spectrum for Antichains of Subsets

Extending a classical theorem of Sperner, we characterize the integers m such that there exists a maximal antichain of size m in the Boolean lattice B_n, that is, the power set of [n] : = { 1 , 2 , … , n} , ordered by inclusion. As an important ingredient in the proof, we initiate the study of an extension of the Kruskal-Katona theorem which is of independent interest. For given positive integers t and k, we ask which integers s have the property that there exists a family F of k-sets with | F| = t such that the shadow of F has size s, where the shadow of F is the collection of (k − 1)-sets that are contained in at least one member of F. We provide a complete answer for t≤ k+ 1. Moreover, we prove that the largest integer which is not the shadow size of any family of k-sets is 2k3/2+84k5/4+O(k).