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Claim. Baumgartner's theorem (§3 of the chapter, as its zbMATH review reports it) states that Martin's axiom for $\aleph_1$ dense sets, $\mathrm{MA}_{\aleph_1}$, makes $\omega_1\omega$ a partition ordinal: for every finite $n$,

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

The second question of Problem 597 follows in every model of $\mathrm{MA}_{\aleph_1}$. A finite graph $G$ on $n$ vertices is a subgraph of $K_n$, so a set of $n$ vertices with all its pairs in the second color contains a copy of $G$; and $\omega_1\omega$ is an initial segment of $\omega_1^2$, so restricting a coloring of $[\omega_1^2]^2$ to $[\omega_1\omega]^2$ transfers the relation. Hence, under $\mathrm{MA}_{\aleph_1}$,

$$\omega_1^2\to(\omega_1\omega,G)^2$$

for every finite $G$, whether or not $G$ is $K_4$-free. Since $\mathrm{MA}_{\aleph_1}$ is consistent relative to ZFC (Solovay and Tennenbaum, Ann. of Math. 94, 1971), the relation for every finite $G$ holds in a model of ZFC, and ZFC does not refute the finite question. The two steps from the theorem to the relation are author-recorded on the result page of the source card, not independently reviewed.

Covers. The second question, the relation for finite $G$: ZFC does not refute a positive answer. It says nothing about the first question, the infinite targets, since the theorem is stated for finite $n$ only, and nothing about the finite question in ZFC, where the triangle case is the theorem of Erdős and Hajnal and every larger case is open. The hypothesis cannot simply be dropped: the continuum hypothesis gives $\omega_1\omega\not\to(\omega_1\omega,3)^2$ (Erdős and Hajnal, as the chapter's review records), although $\omega_1^2\to(\omega_1\omega,3)^2$ is a theorem of ZFC, so a negative relation on $\omega_1\omega$ does not pass to $\omega_1^2$.

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, and the chapter's theorem numbering is unknown. The volume carries only the year, so this page is dated the first of January 1989.

Acceptance. None, so the claim stays claimed. The site labels Problem 597 OPEN and its commentary does not mention the theorem; the chapter appeared in a Springer Lecture Notes in Mathematics proceedings volume, and there is no evidence that the volume's chapters were refereed, so refereed is not listed; and the same chapter is accepted on Problem 1171's page only through that problem's own NOT DISPROVABLE label, whose credit to Baumgartner covers the case $n=3$ alone and says nothing about Problem 597.

Depends on. The main theorem page of the source card, a library result page, which states the theorem and the restriction step.