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Claim. Theorem 3.1 of the deposit states that Martin's axiom for $\aleph_1$ dense sets, $\mathrm{MA}_{\aleph_1}$, implies

$$\omega_1^2\to(\omega_1\omega,\underbrace{3,\ldots,3}_{k})^2_{k+1}$$

for every finite $k\ge1$, the exact relation of Problem 1171 under an added hypothesis; the abstract calls this a conditional affirmative answer. The proof is a color-reduction lemma (Lemma 2.1): an ordinal $\alpha$ with $\alpha\to(\alpha,3)^2$ satisfies $\alpha\to(\alpha,3,\ldots,3)^2_{k+1}$ with $k$ triangle targets for every finite $k\ge1$, by merging two colors and inducting on $k$; it is applied to $\alpha=\omega_1\omega$, which satisfies $\omega_1\omega\to(\omega_1\omega,3)^2$ under $\mathrm{MA}_{\aleph_1}$ by Baumgartner's theorem, and a coloring of $[\omega_1^2]^2$ is restricted to the initial segment $\omega_1\omega$. The hypothesis $\mathrm{MA}_{\aleph_1}$ is not a theorem of ZFC and fails under CH, which gives $\omega_1\omega\not\to(\omega_1\omega,3)^2$. With the Solovay--Tennenbaum consistency of $\mathrm{MA}_{\aleph_1}$ relative to ZFC, the theorem gives a model of ZFC in which the relation holds for every $k$, so ZFC does not refute it: the outcome that a comment of the claimant in the problem's discussion thread (06:50 on 5 September 2026) asked the site to label NOT DISPROVABLE. The ZFC question for $k\ge3$ is untouched.

Covers. One side of an independence result: under the added hypothesis the relation holds for every finite $k$, which with the consistency of $\mathrm{MA}_{\aleph_1}$ means that ZFC does not refute it. Whether ZFC proves the relation is not addressed, so the claim leaves the problem open.

Source and standing. Lezhe Gao, A finite-color partition relation for $\omega_1^2$ under $\mathrm{MA}_{\aleph_1}$, Zenodo deposit published 5 September 2026, version 1, four pages, concept DOI 10.5281/zenodo.22315956, version DOI 10.5281/zenodo.22315957; not refereed and not on arXiv. The site's proof-claims tab listed it as a partial claim submitted 2026-09-05 (partial because the ZFC question stays open), with no comments, under the forum name lezhe; the entry named the AI systems used, but their names are not on record and the entry has been removed, so this page cannot give them. The deposit's Lemma 2.1 and Theorem 3.1 are compiled with their proofs on the source card and reconstructed in the research folder for this problem; an independent review of 2026-09-28 graded the reconstructions faithful and sound, with no tier (the grade); no one has refereed the result.

Withdrawal. The author withdrew the deposit. Its Zenodo record (version DOI 10.5281/zenodo.22315957, to which the concept DOI resolves) is a tombstone. It says the owner removed the record on 1 October 2026 and gives retraction or withdrawal of the record as the reason. The site's proof-claims tab then dropped the claim. The problem's standing takes nothing from this page: the consistency of the relation rests on Baumgartner's theorem, whose full form, $\omega_1\omega\to(\omega_1\omega,n)^2$ for every finite $n$, gives the same conclusion through the finite Ramsey theorem.

Depends on. Baumgartner 1989 for $\omega_1\omega\to(\omega_1\omega,3)^2$ under $\mathrm{MA}_{\aleph_1}$, the input the deposit imports (main theorem).