The strong tree property and weak square

TL;DR

This paper demonstrates the consistency of the super tree property at all finite levels below , while also allowing weak square and good scales at , using large cardinal assumptions.

Contribution

It establishes the simultaneous consistency of the super tree property at all  levels and the existence of weak square and good scales at .

Findings

Super tree property holds at  for all finite n.

Weak square and good scale exist at .

Consistency results rely on  many supercompact cardinals.

Abstract

We show that it is consistent, relative to $\omega$ many supercompact cardinals, that the super tree property holds at $\aleph_n$ for all $2 \leq n < \omega$ but there are weak square and a very good scale at $\aleph_{\omega}$.