Singular Cardinal Combinatorics and Inner Model Theory

These are the slides of a talk given at the Singular Cardinal Combinatorics and Inner Model Theory conference (Gainesville, 5–9 March 2007).

Talk Title: Antichains in partially ordered sets of singular cofinality

Abstract: We say that a singular cardinal $\lambda$ is a prevalent singular cardinal iff there exists a family $\mathcal{F}$ of size $\lambda$ with $\sup\{ |A| : A\in\mathcal{F}\}<\lambda$ such that any subset of $\lambda$ of size less than $\text{cf}(\lambda)$ is covered by some element of $\mathcal F$.

In their paper from 1981, Milner and Sauer conjectured that any poset $\mathbb P$ of singular cofinality, must contain an antichain of size $\text{cf}(\text{cf}(\mathbb{P}))$.

We prove their conjecture restricted to the class of all prevalent singular cardinals.

It is an open problem whether there consistently exists a singular cardinal which is not a prevalent singular cardinal.

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