A variant of Shelah's characterization of Strong Chang's Conjecture

Abstract

Shelah considered a certain version of Strong Chang's Conjecture, which we denote ${\text{SCC}}^{\text{cof}}$, and proved that it is equivalent to several statements, including the assertion that Namba forcing is semiproper. We introduce an apparently weaker version, denoted ${\text{SCC}}^{\text{split}}$, and prove an analogous characterization of it. In particular, ${\text{SCC}}^{\text{split}}$ is equivalent to the assertion that the the Friedman-Krueger poset is semiproper. This strengthens and sharpens the results of Cox, and sheds some light on problems from Usuba and Torres-Perez and Wu.