Polarized relations at singulars over successors

Erdos, Hajnal and Rado asked whether $\binom{\aleph_{ω_1}}{\aleph_2}\rightarrow\binom{\aleph_{ω_1}}{\aleph_0}_2$ and whether $\binom{\aleph_{ω_1}}{\aleph_2}\rightarrow\binom{\aleph_{ω_1}}{\aleph_1}_2$. We prove that both relations are independent over ZFC. We shall also prove that $\binomμ{\aleph_2}\rightarrow\binomμ{\aleph_2}_2$ is independent over ZF for some $μ$ of cofinality $ω_1$.

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