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Claim. Corollary 7 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), stated in §15.6 (p. 132) as one of the three propositions behind part (i) of the first Main Theorem, carries the mark $(*)$ that the paper prefixes to results whose proofs assume the General Continuum Hypothesis (pp. 96--97). For $r=2$ it states $\aleph_\beta\not\to(\aleph_\beta,(3)_{\aleph_{\mathrm{cr}(\beta)}})^2$ for every ordinal $\beta$, where $\mathrm{cr}(\beta)$ is the paper's critical number (§15.1, p. 130); when $\beta=\alpha+1$, $\aleph_{\mathrm{cr}(\beta)}=\mathrm{cf}(\aleph_\alpha)$ (pp. 96 and 130). For $\beta=\omega+1$ this gives $\aleph_{\mathrm{cr}(\omega+1)}=\mathrm{cf}(\aleph_\omega)=\aleph_0$, and the statement is the relation of Problem 1168,

$$\aleph_{\omega+1}\not\to(\aleph_{\omega+1},(3)_{\aleph_0})^2,$$

the case written as display (11) in the proof (p. 133). GCH holds in Gödel's constructible universe, so if ZFC is consistent, so is ZFC with the relation, and ZFC does not refute it. The specialization to $\beta=\omega+1$ is this page's.

Covers. The not-disprovable side of Problem 1168, relative to the consistency of ZFC: the relation holds in every model of ZFC with GCH. It does not prove the relation in ZFC, which is what the problem asks, and one side alone leaves the question open. Garti, Hayut and Shelah (card) restate the result as their Theorem 0.1, for singular $\lambda$ with $2^\lambda=\lambda^+$, and force the relation at $\aleph_{\omega+1}$ together with $2^{\aleph_\omega}>\aleph_{\omega+1}$; the problem page records that work as context.

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. erdosproblems.com credits the result to the paper, under GCH, on a problem it labeled OPEN, so no curator acceptance is listed. The proof of the case $\beta=\alpha+1$ with $\aleph_\alpha$ singular (Case 1b, p. 133) cites Theorem 10, which is printed (p. 117) for $a>a'>\aleph_0$, where $a'$ is the cofinality of $a$; at $\alpha=\omega$ the cardinal $a=\aleph_\omega$ has $a'=\aleph_0$, outside that printed hypothesis. This corpus has not checked whether the proof of Theorem 10 covers countable cofinality, and the proofs are not reviewed here. The result does not rest on that printed route alone: Garti, Hayut and Shelah (card; Theorem 0.1, p. 2) state it for every singular $\lambda$ with $2^\lambda=\lambda^+$, countable cofinality included, as phrased and proved in the 1984 monograph of Erdős, Hajnal, Máté and Rado, Combinatorial set theory: partition relations for cardinals (North-Holland), whose Theorem 20.2 they cite as a stronger form. The monograph is not read here.

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