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 √(2)k^3/2+√(8)k^5/4+O(k) .