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Claim. Theorem I of P. Erdős, A. Hajnal and R. Rado, Partition relations for cardinal numbers, Acta Math. Acad. Sci. Hungar. 16 (1965), no. 1--2, 93--196 (the corpus's source card), the paper's first Main Theorem (§15.2, p. 130), carries the mark $(*)$ that the paper prefixes to results whose proofs assume the General Continuum Hypothesis $2^{\aleph_\nu}=\aleph_{\nu+1}$ (pp. 96--97). Its part (iv), for pairs and two colors, states: if $b$ is a cardinal with $2<b\le\aleph_\beta$, $b<\mathrm{cf}(\aleph_\beta)$ and $b\le\aleph_{\mathrm{cr}(\beta)}$, then $\aleph_\beta\to(\aleph_\beta,b)^2$. Here $\mathrm{cr}(\beta)$ is the paper's critical number (§15.1, p. 130): $\aleph_{\mathrm{cr}(\beta)}$ is the cofinality of $\nu$ when $\mathrm{cf}(\aleph_\beta)=\nu^+$, and is $\mathrm{cf}(\aleph_\beta)$ itself when that is a limit cardinal. The proof of part (iv) (§15.8, p. 135) passes to the cofinality by Lemma 4 (§10.1, p. 113), which under GCH makes $a\to(a,b)^2$ equivalent to $\mathrm{cf}(a)\to(\mathrm{cf}(a),b)^2$, and proves the relation there by Corollary 1 (p. 105) for a successor cofinality and by Theorem 5 (p. 108) for an inaccessible one. Let $\lambda=\aleph_\beta$ be a singular cardinal such that $\lambda$ and $\mathrm{cf}(\lambda)$ are both $\aleph_0$-inaccessible (each exceeds $\mu^{\aleph_0}$ for every smaller $\mu$), as in the Statement of Problem 1220, and take $b=\aleph_1$. Then $\aleph_1\le2^{\aleph_0}<\mathrm{cf}(\lambda)$. If $\mathrm{cf}(\lambda)=\nu^+$, then $\nu^{\aleph_0}<\nu^+$ gives $\nu^{\aleph_0}=\nu$, so $\mathrm{cf}(\nu)>\aleph_0$ by König's theorem and $\aleph_{\mathrm{cr}(\beta)}=\mathrm{cf}(\nu)\ge\aleph_1$; if $\mathrm{cf}(\lambda)$ is a limit cardinal, $\aleph_{\mathrm{cr}(\beta)}=\mathrm{cf}(\lambda)>\aleph_1$. So under GCH part (iv) gives $\lambda\to(\lambda,\aleph_1)^2$ for every such $\lambda$: the universal statement holds in every model of ZFC with GCH. GCH holds in Gödel's constructible universe, so if ZFC is consistent, so is ZFC with the universal statement, and ZFC does not refute it. The specialization to the problem's hypotheses is this page's.

Covers. The not-disprovable side of Problem 1220, relative to the consistency of ZFC: ZFC does not refute that every singular $\lambda$ with $\lambda$ and $\mathrm{cf}(\lambda)$ both $\aleph_0$-inaccessible satisfies $\lambda\to(\lambda,\aleph_1)^2$. It settles that side only; it does not show that ZFC fails to prove the statement, and one side alone leaves the question open. The not-provable side is on Shelah and Stanley's page, and the two pages together settle the problem as independent of ZFC.

Acceptance. Refereed: the result is a journal paper in the Acta Mathematica Academiae Scientiarum Hungaricae, volume 16, issue 1--2. The issue is dated March 1965 and carries no day, so this page's date is the first of that month. The site labels the problem OPEN, so no curator acceptance is listed. This page states Theorem I and the steps of the proof of its part (iv) as the published paper prints them; the proofs of Lemma 4, Corollary 1 and Theorem 5 are not reviewed in this corpus.

Depends on. No other wiki page; the claim rests on the paper above.