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

TL;DR

The paper proves that the existence of an rd
51s cardinal implies the consistency of Chang's Conjecture with a specific square principle, answering questions posed by Sakai and establishing optimality.

Contribution

It demonstrates that an rd
51s cardinal suffices for certain combinatorial set theory principles, extending understanding of their consistency and compatibility.

Findings

Existence of an rd
51s cardinal implies consistency of Chang's Conjecture with -square.

This result is optimal, as shown by Donder's work.

Provides answers to Sakai's questions on incompatibility of certain square principles and Chang's Conjecture.

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 $\kappa$.