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 $\text{cov}\left(\lambda,\mu,\text{cf}(\lambda),2\right)=\lambda$, 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 PFAGCH, and SSH.
In fact, the consistency of the existence of a singular cardinal $\lambda$ satisfying $\left(\lambda,\mu,\text{cf}(\lambda),2\right)>\lambda$ for all $\mu<\lambda$ is unknown.


[No preprint]

Citation information:

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

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