A coloring theorem for succesors of singular cardinal

TL;DR

The paper proves a strong coloring theorem at successors of singular cardinals within ZFC and applies it to address questions about Shelah's principle for such cardinals.

Contribution

It introduces a new coloring theorem valid at successors of singular cardinals and uses it to resolve open questions related to Shelah's principle.

Findings

Established a strong coloring theorem in ZFC for successors of singular cardinals.

Answered several open questions about Shelah's principle $Pr_1$ at singular cardinals.

Provided new tools for analyzing combinatorial properties of singular cardinals.

Abstract

We formulate and prove (in {\sf ZFC}) a strong coloring theorem which holds at successors of singular cardinals, and use it to answer several questions concerning Shelah's principle $Pr_1(\mu^+,\mu^+,\mu^+,\cf(\mu))$ for singular $\mu$.