Failures of square in Pmax extensions of Chang models

TL;DR

This paper demonstrates that certain combinatorial principles, specifically square principles at  and , fail in the -max extension of a modified Chang model, highlighting limitations of this extension.

Contribution

It shows the failure of  and  square principles in -max extensions of a specific Chang model, revealing new limitations of these models.

Findings

 and  square principles fail in the -max extension

Failure occurs in a variation of the Chang model by Sargsyan

Results highlight limitations of -max extensions for certain combinatorial statements

Abstract

We show that the statements $\square(\omega_{3})$ and $\square(\omega_{4})$ both fail in the $\mathbb{P}_{\mathrm{max}}$ extension of a variation of the Chang model introduced by Sargsyan.