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Claim. The answer to Problem 118 is no. In arrow notation the question asks whether $\alpha\to(\alpha,3)^2$ forces $\alpha\to(\alpha,n)^2$ for every finite $n$. Schipperus proves, for every countable $\beta$ that is the sum of one or two indecomposable ordinals, that $\omega^{\omega^\beta}\to(\omega^{\omega^\beta},3)^2$ (Theorem 28, p. 1212), and, for every $\beta$ that is the sum of exactly two indecomposables, that $\omega^{\omega^\beta}\not\to(\omega^{\omega^\beta},6)^2$ (Theorem 29(1), p. 1213, proved as Theorem 31, p. 1214). Since $2=1+1$ is the sum of two indecomposables, $\alpha=\omega^{\omega^2}$ satisfies the hypothesis of the problem, every two-coloring of $K_\alpha$ containing a red $K_\alpha$ or a blue $K_3$, while some two-coloring of $K_\alpha$ contains neither a red $K_\alpha$ nor a blue $K_6$. The paper draws this conclusion itself in its closing remark (p. 1215) and announces it in the abstract as the example that $\alpha\to(\alpha,3)^2$ need not give $\alpha\to(\alpha,n)^2$ for all finite $n$.

Proof shape. The positive relation represents $\omega^{\omega^\beta}$ by finite labeled trees, plays a game in which a Builder builds pairs of trees and an Architect restricts the Builder's moves, and applies a Ramsey dichotomy from the Nash-Williams theorem: either the Architect has a winning strategy and three trees pairwise in color 1 are built, or every sufficiently large play of the Builder wins and a homogeneous set of order type $\omega^{\omega^\beta}$ in color 0 is extracted. The negative relation colors a pair of trees by an interlacing pattern that occurs in every set of the full order type and that no six trees can pairwise exhibit. The statements are recorded on the library pages Theorem 28 and Theorem 29; the source card rests on the one-paragraph proof of Theorem 28 and the pattern arguments of Theorems 31--33, and covers their supporting sections for structure only. Nothing on this page is independently reviewed by this project.

Related results. The paper says (p. 1197) that the negative relations for finite $\beta$ were found independently by Darby, whose paper has its own claim page, Darby 1999, and that Darby also proved the positive relation at $\beta=2$ independently. It also reports, without proof, that Larson found the exact boundary at $\beta=2$: $\omega^{\omega^2}\to(\omega^{\omega^2},4)^2$ but $\omega^{\omega^2}\not\to(\omega^{\omega^2},5)^2$ (the problem page's [La00], with its own claim page, Larson 2000), so the smallest $n$ at which the question fails for this $\alpha$ is $5$. Chapter 2.9 of the Handbook of Set Theory [HST10] gives the background and proof sketches.

Source. Rene Schipperus, Countable partition ordinals, Ann. Pure Appl. Logic 161 (2010), 1195--1215, doi:10.1016/j.apal.2009.12.007; received 9 May 2007, accepted 26 December 2009, available online 13 May 2010. The result was first written up in the author's 1999 thesis of the same title ([Sc99] on the problem page, 57 pages, not held here), which the site credits as the result's first appearance; the thesis carries no day, so this page is dated by the first day of its year, and the labels used here are the journal version's.

Acceptance. Refereed: the result is a journal paper in Annals of Pure and Applied Logic, communicated by T. Jech. Reviewed: the curator of erdosproblems.com, T. F. Bloom, marks Problem 118 disproved and credits Schipperus's thesis and its published version, together with Darby, as the independent disproofs (problem page last edited 17 January 2026).