Unbalanced polarized relations

TL;DR

This paper proves the consistency of a polarized partition relation involving a strong limit singular cardinal with countable cofinality, demonstrating its validity at various large and small cardinal levels.

Contribution

It establishes the consistency of a specific polarized partition relation at singular and measurable cardinals, extending known results in set theory.

Findings

The relation can be forced at limit of measurable cardinals.

The relation holds at small cardinals like leph_.

The proof involves advanced forcing techniques.

Abstract

We prove the consistency of $\binom{\mu^+}{\mu}\nrightarrow\binom{\mu^+ \omega_1}{\mu\ \mu}$ where $\mu$ is a strong limit singular cardinal of countable cofinality. This result can be forced at limit of measurable cardinals and at small cardinals like $\aleph_\omega$.