# Blog Archives

# Topic Archive: weakly compact cardinals

Set theory seminarFriday, October 16, 201510:00 amGC 3212

# Layered partial orders

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.