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Claim. Galvin and Larson's Theorem 9 reads: "If $\alpha<\omega_1$ and $\alpha\to(\alpha,3)^2$, then, either $\alpha\in\{0,1,\omega^2\}$, or else $\alpha=\omega^{\omega^\beta}$ for some $\beta<\omega_1$." In the reading $\alpha=\omega^\beta$ of Problem 592 (its Formulation), it follows that $\omega^\beta\not\to(\omega^\beta,3)^2$ for every decomposable $\beta$ with $3\le\beta<\omega_1$, so no such $\beta$ has the property, and that the question reduces to the exponents $\beta=\omega^\gamma$. The proof combines the paper's Theorem 3, that $\omega^\varepsilon$ can be pinned to $\omega^3$ whenever $\varepsilon$ is decomposable and $3\le\varepsilon<\omega_1$ (proved through Lemmas 4--6, Lemma 4 being Specker's), with Specker's $\omega^3\not\to(\omega^3,3)^2$ and his observation that a partition relation passes along a pinning map. Theorem 2 of the paper adds the converse, that $\omega^\varepsilon$ cannot be pinned to $\omega^3$ when $\varepsilon$ is indecomposable (Theorem 8), so pinning gives no negative relation at the exponents left open. The paper is F. Galvin and J. Larson, Pinning countable ordinals, Fund. Math. 82 (1974/75), no. 4, 357--361, DOI 10.4064/fm-82-4-357-361, the site's [GaLa74], received 10 September 1973; the publisher's record gives the year 1975 and no month or day, so the page name carries 1 January 1975. The paper is filed with a transcription on the library's source card.

Covers. Every decomposable exponent $\beta$ with $3\le\beta<\omega_1$: the property fails. This contains Specker's finite $\beta\ge3$. Not covered: the exponents $\beta\le2$ and the indecomposable $\beta=\omega^\gamma$, among them Chang's $\beta=\omega$ and the cases that Schipperus decides.

Depends on. Specker's relation $\omega^3\not\to(\omega^3,3)^2$ and the transfer of partition relations along pinning maps, which the paper's proof of Theorem 9 cites.

Acceptance. Refereed: the paper appeared in Fundamenta Mathematicae, volume 82. The site labels the problem OPEN, and its commentary crediting Galvin and Larson on an open problem is not an acceptance, so no reviewed evidence is listed.