Unbalanced polarized relations

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 ${\mathrm{\aleph }}_{\omega }$.