Erdős #1220 · set theory / ramsey theory

A problem of Erdős, Hajnal, and Rado, who note that the answer is positive if we assume GCH or 'some other additional assumptions' on $\lambda$.

We say a cardinal $\kappa$ is $\aleph_0$-inaccessible if for all $\mu<\kappa$ we have $\mu^{\aleph_0}<\kappa$. These conditions are necessary, since $\tau^{\aleph_0}\not\to (\tau^+,\aleph_1)^2$ and if $\mathrm{cf}(\kappa)\not\to (\mathrm{cf}(\kappa),\aleph_1)^2$ then $\kappa\not\to (\kappa,\aleph_1)^2$.

The simplest case is $\lambda=\aleph_{\mathfrak{c}^+}$, in which case the second condition implies the first by a theorem of Shelah [Sh82].

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