# On the consistency strength of the Milner-Sauer conjecture

Abstract: In their paper from 1981, Milner and Sauer conjectured that for any poset $\mathbb P$, if $\text{cf}(\mathbb P)$ is a singular cardinal $\lambda$, then $\mathbb P$ must contain an antichain of size $\text{cf}(\lambda)$.

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 cardinals $\lambda$ above the first strongly compact cardinal.