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Claim. Erdős and Rado prove, for every infinite cardinal $\kappa$, that $(2^\kappa)^+\to(\kappa^+)^2_\kappa$: every coloring of the pairs of a set of size $(2^\kappa)^+$ with $\kappa$ colors has a homogeneous set of size $\kappa^+$. The site's remark quotes the stronger ordinal form $(2^\kappa)^+\to(\kappa^++1)^2_\kappa$; the cardinal form suffices here.

The step to the first relation of Problem 1172 is one line (author-recorded). Under GCH, $2^{\aleph_1}=\aleph_2$, so the theorem at $\kappa=\aleph_1$ gives $\omega_3\to(\aleph_2)^2_{\aleph_1}$, and in particular every $2$-coloring of $[\omega_3]^2$ has a homogeneous set of size $\aleph_2$. A set of ordinals of size $\aleph_2$ has order type at least $\omega_2$, so it contains subsets of types $\omega_2$ and $\omega_1+2$. Whichever color the homogeneous set has, the relation $\omega_3\to(\omega_2,\omega_1+2)^2$ holds.

Covers. The first relation, $\omega_3\to(\omega_2,\omega_1+2)^2$ under GCH, as the site prints it. The problem page's Formulation records that the booklet's version of this relation is cut off, so the printed target may not be the one Erdős and Hajnal intended. The second and third relations and the consistency question are not covered.

Depends on. No page of this wiki.

Source. P. Erdős and R. Rado, A partition calculus in set theory, Bull. Amer. Math. Soc. 62 (1956), no. 5, 427-489, doi:10.1090/S0002-9904-1956-10036-0 (source card). The issue is dated September 1956 and the record carries no day, so this page's date is the first of that month.

Acceptance. Refereed: the theorem is in a journal paper in the Bulletin of the American Mathematical Society.