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Claim. Schipperus's Theorem 28 (p. 1212) reads: "Let $\beta<\omega_1$ be the sum of one or two indecomposable ordinals, then $\omega^{\omega^\beta}\to(\omega^{\omega^\beta},3)^2$." Part 3 of Schipperus's Theorem 29 (p. 1213) reads: "If $\beta$ is the sum of $\ge4$ indecomposable ordinals then $\omega^{\omega^\beta}\not\to(\omega^{\omega^\beta},3)^2$." The paper's $\beta$ is the $\gamma$ of Problem 592 in its reading $\alpha=\omega^\beta$ (the Formulation), with the problem's exponent $\beta=\omega^\gamma$. So the exponent $\beta=\omega^\gamma$ has the property when $\gamma$ is the sum of one or two indecomposable ordinals and lacks it when $\gamma$ is the sum of four or more. Theorem 28 is proved on p. 1212 from the paper's Ramsey dichotomy for the Builder--Architect game (Theorem 19), its homogeneous-set theorem (Theorem 21) and its triangle lemma (Lemma 26), after a reduction by the Erdős--Milner theorem (Theorem 27, cited to Williams's Combinatorial Set Theory); Theorem 29 is proved as Theorems 31--33 (pp. 1214--1215) by colorings that record a pattern of interlacing. The paper is R. Schipperus, Countable partition ordinals, Ann. Pure Appl. Logic 161 (2010), 1195--1215, DOI 10.1016/j.apal.2009.12.007, the site's [Sc10], received 9 May 2007, accepted 26 December 2009 and available online 13 May 2010 (p. 1195), the date the page name carries. It is paged on the library's source card.

Covers. The exponents $\beta=\omega^\gamma$ with $\gamma$ the sum of one or two indecomposable ordinals (the property holds; $\gamma=1$ is Chang's $\beta=\omega$) and with $\gamma$ the sum of four or more (it fails). Not covered: $\gamma$ the sum of exactly three indecomposable ordinals, where Theorem 29 gives only $\omega^{\omega^\gamma}\not\to(\omega^{\omega^\gamma},4)^2$ and the relation with $3$ is undecided.

Depends on. No page of this wiki.

Acceptance. Refereed: the paper appeared in the Annals of Pure and Applied Logic, volume 161. The site labels the problem OPEN, and its commentary crediting Schipperus on an open problem is not an acceptance, so no reviewed evidence is listed.

Read depth. The statements of Theorems 28 and 29 and the one-paragraph proof of Theorem 28 are checked; the sections that the proof of Theorem 28 assembles (§§ 2--10) were read for structure only, and the proofs of Theorems 31--33 were not checked.