On the consistency of local and global versions of Chang's Conjecture

TL;DR

This paper investigates the consistency of certain instances of Chang's Conjecture involving pairs of infinite cardinals, establishing their consistency relative to large cardinal assumptions.

Contribution

It proves the relative consistency of specific forms of Chang's Conjecture for many pairs of cardinals, extending previous results and answering open questions.

Findings

Consistency of Chang's Conjecture for many pairs of cardinals under supercompact assumptions.

Consistency of Chang's Conjecture for all successor cardinals under huge cardinal assumptions.

Addresses an open question by Foreman regarding the conjecture's global versions.

Abstract

We show that for many pairs of infinite cardinals $\kappa > \mu^+ > \mu$, $(\kappa^{+}, \kappa)\twoheadrightarrow (\mu^+, \mu)$ is consistent relative to the consistency of a supercompact cardinal. We also show that it is consistent, relative to a huge cardinal that $(\kappa^{+}, \kappa)\twoheadrightarrow (\mu^+, \mu)$ for every successor cardinal $\kappa$ and every $\mu < \kappa$, answering a question of Foreman.