Tree property at successor of a singular limit of measurable cardinals

TL;DR

This paper introduces two forcing techniques to establish the tree property at the successor of a singular limit of measurable cardinals, extending previous results and addressing the failure of the SCH at certain cardinals.

Contribution

It presents novel forcing methods to achieve the tree property at successors of singular cardinals under specific large cardinal assumptions, extending prior work by Neeman, Sinapova, Magidor, and Shelah.

Findings

Tree property at a^+ for a a limit of supercompact cardinals

Failure of SCH at a_{a^2} with tree property at its successor

Extension of previous results to arbitrary singular cardinals

Abstract

Assume $\lambda$ is a singular limit of $\eta$ supercompact cardinals, where $\eta \leq \lambda$ is a limit ordinal. We present two forcing methods for making $\lambda^+$ the successor of the limit of the first $\eta$ measurable cardinals while the tree property holding at $\lambda^+.$ The first method is then used to get, from the same assumptions, tree property at $\aleph_{\eta^2+1}$ with the failure of $SCH$ at $\aleph_{\eta^2}$. This extends results of Neeman and Sinapova. The second method is also used to get tree property at successor of an arbitrary singular cardinal, which extends some results of Magidor-Shelah, Neeman and Sinapova.