Strong tree Properties for two successive cardinals

Abstract

An inaccessible cardinal $\kappa$ is supercompact when $(\kappa ,\lambda )$-ITP holds for all $\lambda \ge \kappa .$ We prove that if there is a model of $\ZFC$ with two supercompact cardinals, then there is a model of \ZFC where simultaneously $({\aleph }_2,\mu )$-ITP and $({\aleph }_3,{\mu }^{\prime })$-ITP hold, for all $\mu \ge {\aleph }_2$ and ${\mu }^{\prime }\ge {\aleph }_3.$