Aronszajn trees, square principles, and stationary reflection

We investigate questions involving Aronszajn trees, square principles, and stationary reflection. We first consider two strengthenings of $\square(κ)$ introduced by Brodsky and Rinot for the purpose of constructing $κ$-Souslin trees. Answering a question of Rinot, we prove that the weaker of these strengthenings is compatible with stationary reflection at $κ$ but the stronger is not. We then prove that, if $μ$ is a singular cardinal, $\square_μ$ implies the existence of a special $μ^+$-tree with a $\mathrm{cf}(μ)$-ascent path, thus answering a question of Lücke.

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