Getting more colors

TL;DR

This paper proves a new coloring theorem for successors of singular cardinals and characterizes when certain partition relations fail for these cardinals.

Contribution

It establishes a coloring theorem for successors of singular cardinals and characterizes the failure of partition relations in this context.

Findings

Proves a coloring theorem for successors of singular cardinals.

Characterizes when $ ext{mu}^+ rightarrow[ ext{mu}^+]^2_{ ext{mu}^+}$ holds or fails.

Shows the equivalence of the failure of certain partition relations for arbitrarily large $ heta< ext{mu}$.

Abstract

We establish a coloring theorem for successors of a singular cardinals, and use it prove that for any such cardinal $\mu$, we have $\mu^+\nrightarrow[\mu^+]^2_{\mu^+}$ if and only if $\mu^+\nrightarrow[\mu^+]^2_{\theta}$ for arbitrarily large $\theta<\mu$.