Aronszajn trees, square principles, and stationary reflection

Abstract

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 Lücke.