Erdős #1218 · set theory / ramsey theory

A problem of Erdős, Hajnal, and Rado.

In [ErHa71] Erdős and Hajnal describe this as the only problem of this type (concerning generalisations of Ramsey's theorem for cardinal numbers), not involving strongly inaccessible cardinals, that remains unsolved assuming GCH. (It is unlikely they meant just this specific question, but rather that this was representative of an edge case of what was uncovered by existing results, see below.)

In general, Erdős, Hajnal, and Rado [EHR65] conjectured that, if

then

$\beta=\omega_{\omega+1}$ is the smallest ordinal which satisfies this condition.

This represented the main unknown for their methods; they proved (assuming GCH), among many other results, that if $r\geq 2$ then

if and only if $b\leq\aleph_{\mathrm{cr}(\beta)}$, except possibly when either $r\geq 3$, $\beta>\mathrm{cf}(\beta)>\mathrm{cf}(\beta)-1>\mathrm{cr}(\beta)$, and $b$ is finite, or possibly when $\aleph_{\mathrm{cf}(\beta)}$ is inaccessible and $>\aleph_0$.

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