On chains and Sperner k-families in ranked posets

Abstract

A ranked poset P has the Sperner property if the sizes of the largest rank and of the largest antichain in P are equal. A natural strengthening of the Sperner property is condition S: For all k, the set of elements of the k largest ranks in P is a Sperner k-family. P satisfies condition T if for all k there exist disjoint chains in P each of which meets the k largest ranks and which covers the kth largest rank. It is proven here that if P satisfies S, it also satisfies T, and that the converse, although in general false, is true for posets with unimodal Whitney numbers. Conditions S and T and the Sperner property are compared here with two other conditions on posets concerning the existence of certain partitions of P into chains.