On the consistency strength of the Milner-Sauer conjecture

In their paper from 1981, Milner and Sauer conjectured that for any poset ( P, ≤) , if cf ( P, ≤ ) = λ > cf (λ ) = K , then P must contain an antichain of cardinality K . The conjecture is consistent and known to follow from GCH-type assumptions. We prove that the conjecture has large cardinals consistency strength in the sense that its negation implies, for example, the existence of a measurable cardinal in an inner model. We also prove that the conjecture follows from Martin's Maximum and holds for all singular λ above the first strongly compact cardinal.