On chains and Sperner k-families in ranked posets, II

Abstract

A ranked poset P satisfies condition S if for all k the set of elements of the k largest ranks in P is a Sperner k-family. It satisfies condition T if for all k there exist disjoint chains in P which each meet the k largest ranks and which cover the kth largest rank. Griggs employed the theory of saturated partitions to prove that if P satisfies S it also satisfies T, and that the converse is true for posets with unimodal Whitney numbers. Here we present new proofs of these facts which do not require the existence of saturated partitions. The first result is proven with a simple network flow argument and the second is proven directly.