Antichains in partially ordered sets of singular cofinality

Abstract

In their paper from 1981, Milner and Sauer conjectured that for any poset P, if cf(P)=lambda>cf(lambda)=kappa, then P must contain an antichain of size kappa. We prove that for lambda>cf(lambda)=kappa, if there exists a cardinal mu<lambda such that cov(lambda,mu,kappa,2)=lambda, then any poset of cofinality lambda contains lambda^kappa antichains of size kappa. The hypothesis of our theorem is very weak and is a consequence of many well-known axioms such as GCH, SSH and PFA. The consistency of the negation of this hypothesis is unknown.