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

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.

Fixed a proof for Lemma 42

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