Polarized relations at singulars over successors

Abstract

Erdos, Hajnal and Rado asked whether $\left(\genfrac{}{}{0ex}{}{{\aleph }_{{\omega }_1}}{{\aleph }_2}\right)\to {\left(\genfrac{}{}{0ex}{}{{\aleph }_{{\omega }_1}}{{\aleph }_0}\right)}_2$ and whether $\left(\genfrac{}{}{0ex}{}{{\aleph }_{{\omega }_1}}{{\aleph }_2}\right)\to {\left(\genfrac{}{}{0ex}{}{{\aleph }_{{\omega }_1}}{{\aleph }_1}\right)}_2$. We prove that both relations are independent over ZFC. We shall also prove that $\left(\genfrac{}{}{0ex}{}{\mu }{{\aleph }_2}\right)\to {\left(\genfrac{}{}{0ex}{}{\mu }{{\aleph }_2}\right)}_2$ is independent over ZF for some $\mu$ of cofinality ${\omega }_1$.

17 pages, to appear