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

Abstract

We show that for many pairs of infinite cardinals κ > μ + > μ \kappa > \mu ^+ > \mu , ( κ + , κ ) ↠ ( μ + , μ ) (\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 ( κ + , κ ) ↠ ( μ + , μ ) (\kappa ^{+}, \kappa )\twoheadrightarrow (\mu ^+, \mu ) for every successor cardinal κ \kappa and every μ > κ \mu > \kappa , answering a question of Foreman.