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Claim. Problem 1171 asks whether $\omega_1^2\to(\omega_1\omega,3,\ldots,3)^2_{k+1}$, with $k$ triangle targets, holds for every finite $k$. Baumgartner's theorem (§3 of the chapter) states that Martin's axiom for $\aleph_1$ dense sets, $\mathrm{MA}_{\aleph_1}$, makes $\omega_1\omega$ and $\omega_1\omega^2$ partition ordinals: for every finite $n$,

$$\omega_1\omega\to(\omega_1\omega,n)^2 .$$

The catalog relation follows. Fix $k\ge1$ and let $n$ be the finite Ramsey number for triangles in $k$ colors. Given a coloring of $[\omega_1^2]^2$ with the colors $0,\ldots,k$, restrict it to the initial segment $\omega_1\omega$ and merge the colors $1,\ldots,k$; the theorem gives a set of type $\omega_1\omega$ homogeneous in color $0$ or an $n$-element set colored from $1,\ldots,k$ only, which contains a monochromatic triangle. So $\mathrm{MA}_{\aleph_1}$ implies the relation for every finite $k$ (the case $k=0$ has one color and is trivial). Since $\mathrm{MA}_{\aleph_1}$ is consistent relative to ZFC (Solovay and Tennenbaum, Ann. of Math. 94, 1971), the relation holds in a model of ZFC and ZFC does not refute it: the relation is not disprovable in the site's sense.

Covers. One side of an independence result: the relation holds in a model of ZFC for every finite $k$, so ZFC does not refute it. The other side, that ZFC does not prove the relation, is not addressed; the relation is a theorem of ZFC for $k\le2$ and open in ZFC for $k\ge3$, so the claim leaves the problem open.

The step from the theorem to the relation. The chapter does not state the catalog relation. The finite Ramsey step above is written out on the result page of the source card and is author-recorded there, not independently reviewed. The only other written form found is the color-reduction lemma of Gao's deposit, which reaches the same conclusion from the case $n=3$ alone by an induction on $k$; that deposit has its own page, Gao's conditional proof, recorded as withdrawn, and the standing here takes nothing from it.

Hypothesis and the ZFC question. The theorem needs an axiom beyond ZFC: the continuum hypothesis gives $\omega_1\omega\not\to(\omega_1\omega,3)^2$ (Erdős and Hajnal, as the chapter's review records), so the partition property of $\omega_1\omega$ is independent of ZFC, and under CH the route through $\omega_1\omega$ closes. The catalog relation itself is a theorem of ZFC for $k\le2$ (Baumgartner and Hajnal 1987 for $k=2$, containing $k=1$) and open in ZFC for $k\ge3$, as Komjáth's 2025 survey records (Problem 54 commentary); the problem page states these results with their sources. Nothing here bears on the ZFC question.

Source. James E. Baumgartner, Remarks on partition ordinals, in Set theory and its applications (Toronto, ON, 1987), Lecture Notes in Mathematics 1401, Springer, Berlin, 1989, pp. 5--17; DOI 10.1007/BFb0097328; Zbl 0703.03027; MR 1031762. The chapter is paywalled and not held; its statement is taken from the zbMATH review and from the restatement in the introduction of Chen, Garti and Weinert, as the source card records; the chapter's theorem numbering is unknown and its proof was not read. The volume carries only the year, so this page is dated the first of January 1989.

Acceptance. Reviewed: the curator of erdosproblems.com, T. F. Bloom, labels the problem not disprovable, and the page's one remark credits Baumgartner's chapter with $\omega_1\omega\to(\omega_1\omega,3)^2$ under a form of Martin's axiom (problem page last edited 26 January 2026; the label and the community database changed to not disprovable on 5 September 2026 after a comment in the discussion thread asked for it on the strength of the deposit since removed). The curator's credit to Baumgartner covers only the two-color relation, and the site writes no step from it to the catalog relation; that step is author-recorded here, so reviewed rests on the label and credit alone and carries the acceptance by itself. The chapter appeared in a Springer Lecture Notes in Mathematics proceedings volume, reviewed in zbMATH and Mathematical Reviews; no evidence that the volume's chapters were refereed was found, so refereed is not listed. Nothing on this page is independently reviewed by this project.

Depends on. Baumgartner 1989, main theorem, the chapter's theorem with the finite Ramsey step written out.