We present two applications of forcing with finite sequences of models as side conditions, adding objects of size \. The first involves adding a \ sequence and variants of such sequences. The second involves adding partial weak specializing functions for trees of height \.

Answering a question of Sakai :29–45, 2013), we show that the existence of an \-Erdős cardinal suffices to obtain the consistency of Chang’s Conjecture with \. By a result of Donder, volume 872 of lecture notes in mathematics. Springer, Berlin, pp 55–97, 1981) this is best possible. We also give an answer to another question of Sakai relating to the incompatibility of \ and \ \twoheadrightarrow \) for uncountable \.

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We investigate questions involving Aronszajn trees, square principles, and stationary reflection. We first consider two strengthenings of 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, implies the existence of a special ‐tree with a cf(μ)‐ascent path, thus answering a question of Lücke.