---

Claim. Theorem 28 (p. 1212): if $\beta<\omega_1$ is the sum of one or two indecomposable ordinals, then $\omega^{\omega^\beta}\to(\omega^{\omega^\beta},3)^2$. With $\beta=2=1+1$ this is the question of Problem 591: in every red/blue coloring of the edges of $K_\alpha$ with $\alpha=\omega^{\omega^2}$ there is a red $K_\alpha$ or a blue $K_3$. The one-paragraph proof reduces, by the Erdős–Milner theorem, to colorings that give color 0, the color of the large homogeneous set, to every pair of equal block type in the tree representation $W_\beta$ of $\omega^{\omega^\beta}$, then applies the paper's Ramsey dichotomy for the Builder–Architect game: one branch gives a triangle in the second color, the other a set of order type $\omega^{\omega^\beta}$ homogeneous in the first. The paper's negative results bound the theorem: $\omega^{\omega^2}\not\to(\omega^{\omega^2},6)^2$ (Theorem 29(1)), so the relation does not extend to all finite cliques; the paper reports Larson's improvement of $6$ to $5$ with $\omega^{\omega^2}\to(\omega^{\omega^2},4)^2$ and does not prove it.

Source. R. Schipperus, Countable partition ordinals, Ann. Pure Appl. Logic 161 (2010), no. 10, 1195–1215, received 2007-05-09, revised 2009-01-01, accepted 2009-12-26, available online 2010-05-13, the date of this page; the author's 1999 thesis of the same title ([Sc99] on Problem 118's page) is not held. The statement in its three printed forms (Theorem 1, Theorem 3, Theorem 28) and the proof of Theorem 28 are the basis of this page, recorded on the source card and its theorem page; Sections 2–10 are followed for structure only and nothing is independently reviewed. The paper states (p. 1197) that Darby independently proved the case $\beta=2$, and the site credits Darby beside Schipperus without a reference; no publication of Darby's proof is held or cited by either, so Darby's proof is disclosed here and has no page of its own.

Acceptance. Refereed: Annals of Pure and Applied Logic. Reviewed: the curator of erdosproblems.com (T. F. Bloom) labels the problem PROVED and credits Schipperus [Sc10] and Darby with independent proofs in the problem's commentary. The curator is independent of the author.

Formalization. The formal-conjectures statement file (2026-10-07) marks its statement erdos_591 research solved with answer(True) and no formal-proof link, and the site's label carries no Lean tag; no formalization of the proof is recorded.