Chang's Conjecture with ${\mathrm{\square }}_{{\omega }_1,2}$ from an ${\omega }_1$-Erd\H{o}s Cardinal

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 ${\mathrm{\square }}_{{\omega }_1,2}$. By a result of Donder this is best possible. We also give an answer to another question of Sakai relating to the incompatibility of ${\mathrm{\square }}_{\lambda ,2}$ and $(\lambda^+, \lambda) \twoheadrightarrow (\kappa^+, \kappa)$foruncountable$\kappa$.