Chang's Conjecture with ${\square }_{{\omega }_1,2}$ from an ${\omega }_1$-Erdős Cardinal

Abstract

Answering a question of Sakai, we show that the existence of an ${\omega }_1$-Erdős 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 $\kappa$.