Chang's Conjecture with $\square_{\omega_1, 2}$ from an $\omega_1$-Erd\H{o}s Cardinal

Abstract

Answering a question of Sakai, we show that the existence of an $\omega_1$-Erd\H{o}s cardinal suffices to obtain the consistency of Chang's Conjecture with $\square_{\omega_1, 2}$. By a result of Donder this is best possible. We also give an answer to another question of Sakai relating to the incompatibility of $\square_{\lambda, 2}$ and $(\lambda^+, \lambda) \twoheadrightarrow (\kappa^+, \kappa)$ for uncountable $\kappa$.