# Layered partial orders

## Sean Cox

### Virginia Commonwealth University

If $\kappa$ is a regular uncountable cardinal and $\mathbb{P}$ is a partial order, we say that $\mathbb{P}$ is $\kappa$ stationarily layered iff the set of regular suborders of $\mathbb{P}$ is stationary in $[\mathbb{P}]^{<\kappa}$. This is a strong form of the $\kappa$-chain condition, and in fact implies that $\mathbb{P}$ is $\kappa$-Knaster. I will discuss two recent applications involving layered posets:

(1) a new characterization of weak compactness: a regular $\kappa$ is weakly compact iff every $\kappa$-cc poset is $\kappa$ stationarily layered. This is joint work with Philipp Luecke.

(2) a general theorem about preservation of $\kappa$-cc under universal Kunen-style iterations.

Sean Cox is an assistant professor at Virginia Commonwealth University. His research interests include set theory, mathematical logic, large cardinals, forcing, inner model theory, and applications of set theory to other branches of mathematics. He received his doctorate from University of California, Irvine in 2009 under the supervision of Martin Zeman and subsequently held postdoctoral positions at the Fields Institute for Research in Mathematical Sciences (Canada) and the Institute for Mathematical Logic and Basic Research, University of Münster (Germany).