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Erdos, Hajnal and Rado asked whether$\binom{\aleph_{\omega_1}}{\aleph_2}\rightarrow\binom{\aleph_{\omega_1}}{\aleph_0}_2$and whether$\binom{\aleph_{\omega_1}}{\aleph_2}\rightarrow\binom{\aleph_{\omega_1}}{\aleph_1}_2$.We prove that both relations are independent over ZFC. We shall also prove that$\binom{\mu}{\aleph_2}\rightarrow\binom{\mu}{\aleph_2}_2$ is independent overZF for some $\mu$ of cofinality $\omega_1$.

Polarized relations at singulars over successors