Tree property at successor of a singular limit of measurable 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.