Antichains in partially ordered sets of singular cofinality

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 main result of of this paper reads as follows:

Suppose that $\lambda$ singular cardinal.
If there exists a cardinal $\mu <\lambda$ for which $\text{cov}(\lambda ,\mu ,\text{cf}(\lambda ),2)=\lambda$, then any poset of cofinality $\lambda$ contains ${\lambda }^{\text{cf}(\lambda )}$ many antichains of size $\text{cf}(\lambda )$.

The hypothesis of the above theorem is very weak and is a consequence of many well-known axioms, icluding PFA, GCH, and SSH.
In fact, the consistency of the existence of a singular cardinal $\lambda$ satisfying $(\lambda ,\mu ,\text{cf}(\lambda ),2)>\lambda$ for all $\mu <\lambda$ is unknown.

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Citation information:

A. Rinot, Antichains in partially ordered sets of singular cofinality, Arch. Math. Logic, 46(5-6): 457-464, 2007.