On the consistency strength of the Milner–Sauer conjecture

In their paper from 1981, Milner and Sauer conjectured that for any poset 〈P,≤〉, if , then P must contain an antichain of cardinality κ. 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.