Aronszajn trees, square principles, and stationary reflection

Chris Lambie-Hanson

Published 2016-05-18Version 1

We investigate questions involving Aronszajn trees, square principles, and stationary reflection. We first consider two strengthenings of $\square(\kappa)$ introduced by Brodsky and Rinot for the purpose of constructing $\kappa$-Souslin trees. Answering a question of Rinot, we prove that the weaker of these strengthenings is compatible with stationary reflection at $\kappa$ but the stronger is not. We then prove that, if $\mu$ is a singular cardinal, $\square_\mu$ implies the existence of a special $\mu^+$-tree with a $\mathrm{cf}(\mu)$-ascent path, thus answering a question of L\"{u}cke. We show this result is sharp by proving that $\square_{\mu, 2}$ is compatible with the statement that every tree of height $\mu^+$ with a narrow ascent path has a cofinal branch.