Squares, ascent paths, and chain conditions, part 1

Two topics of interest in modern set theory are the productivity of chain conditions and the existence of higher Aronszajn trees.

In this talk, we discuss generalizations of both of these topics and their connections with various square principles.

In particular, we will prove that, if $\kappa$ is a regular uncountable cardinal and $\square(\kappa)$ holds, then:

1) for all regular $\lambda < \kappa$, there is a $\kappa$-Aronszajn tree with a $\lambda$-ascent path;

2) there is a $\kappa$-Knaster poset $\mathbb{P}$ such that $\mathbb{P}^{\aleph_0}$ is not $\kappa$-c.c. 

Time permitting, we will also present a complete picture of the relationship between the existence of special trees and the existence of Aronszajn trees with ascent paths at the successor of a regular cardinal.

 

This is joint work with Philipp Lücke.