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Claim. The answer to Problem 118 is no: there is an ordinal $\alpha$ with $\alpha\to(\alpha,3)^2$ for which $\alpha\to(\alpha,n)^2$ fails at some finite $n$. The site credits Darby's paper, independently of Schipperus, with this disproof. Schipperus's own account (Countable partition ordinals, p. 1197) is that the negative relations $\omega^{\omega^\beta}\not\to(\omega^{\omega^\beta},6)^2$ for $\beta$ the sum of two indecomposable ordinals, $\not\to(\cdot,4)^2$ for three and $\not\to(\cdot,3)^2$ for four or more are, for finite $\beta$, due independently to Darby and to himself, and that Darby also proved the positive relation $\omega^{\omega^2}\to(\omega^{\omega^2},3)^2$ independently; the two together give the counterexample $\alpha=\omega^{\omega^2}$ at $n=6$. Erdős reported the same result in 1995 (§ 1 of Problems in combinatorial set theory): a letter from Laver had informed him that Darby disproved his conjecture, proving that some countable $\alpha$ satisfies $\alpha\to(\alpha,3)^2$ but not $\alpha\to(\alpha,6)^2$.

Sources. The account rests on the paper's title and journal record, on Schipperus's report (Countable partition ordinals, p. 1197), on Erdős 1995 §1 and on the curator's credit; none of them places the positive relation $\omega^{\omega^2}\to(\omega^{\omega^2},3)^2$ in the JCTB paper, since Schipperus's report of Darby's positive proof cites no paper for it and Erdős's names none. The only published proof of that relation among these sources is Schipperus's Theorem 28, on the companion claim page Schipperus 1999, which records the same counterexample from the journal paper; the exact boundary at $\alpha=\omega^{\omega^2}$, $n=5$, is Larson's, Larson 2000. Nothing on this page is independently reviewed by this project.

Depends on. Schipperus 1999 for the published proof of the positive relation $\omega^{\omega^2}\to(\omega^{\omega^2},3)^2$, which Darby is reported to have proved independently in work not held here.

Source. Carl Darby, Negative partition relations for ordinals $\omega^{\omega^\alpha}$, J. Combin. Theory Ser. B 76 (1999), no. 2, 205--222, doi:10.1006/jctb.1999.1903. The issue is dated July 1999 and carries no day, so this page's date is the first of that month.

Acceptance. Reviewed: the curator of erdosproblems.com, T. F. Bloom, marks Problem 118 disproved and names Darby, with Schipperus, as having shown the answer is no independently (problem page last edited 17 January 2026). The paper appeared in a refereed journal, but refereed is not listed because no source read places the positive half of the counterexample in the paper.