Unbalanced polarized relations

Abstract

We prove the consistency of $\left(\genfrac{}{}{0ex}{}{{\mu }^{+}}{\mu }\right)\nrightarrow \left(\genfrac{}{}{0ex}{}{{\mu }^{+}{\omega }_1}{\mu \mu }\right)$ 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 ${\aleph }_{\omega }$.

Two open problems were added at the end