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(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 cf(mu)-ascent path, thus answering a question of Lucke. (C) 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim