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Claim. As the zbMATH review of the paper (Zbl 0635.03042) states the result, Baumgartner and Hajnal prove that for every regular cardinal $\kappa$ with $\kappa^{<\kappa}=\kappa$

$$(\kappa^+)^2\to(\kappa^+\kappa,3,3)^2;$$

that is, every coloring of the pairs from the ordinal $(\kappa^+)^2$ with three colors has a set of order type $\kappa^+\kappa$ homogeneous in color $0$ or a triangle homogeneous in color $1$ or in color $2$. For $\kappa=\omega$ the hypothesis $\omega^{<\omega}=\omega$ holds in ZFC and $\kappa^+=\omega_1$, so $\omega_1^2\to(\omega_1\omega,3,3)^2$ is a theorem of ZFC: the case $k=2$ of Problem 1171. Leaving one triangle color unused gives the case $k=1$, $\omega_1^2\to(\omega_1\omega,3)^2$. Komjáth's 2025 survey of the Erdős--Hajnal problem list restates the relation for $\kappa=\omega$ (Problem 13 commentary and Problem 54 discussion) and describes its proof as quite complicated. The result is compiled on the library's positive relation page of the source card.

Covers. The instances $k=1$ and $k=2$ of the question, in ZFC. The relation for $k\ge3$ is not addressed; Komjáth records the case $k=3$, $\omega_1^2\to(\omega_1\omega,3,3,3)^2$, as unknown. The same paper shows that the continuum hypothesis gives $\omega_1^2\not\to(\omega_1\omega,4)^2$, so the triangle targets cannot be raised to four; that relation is context, not part of this claim.

Standing. Claimed. James E. Baumgartner and András Hajnal, A remark on partition relations for infinite ordinals with an application to finite combinatorics, in Logic and combinatorics (Arcata, Calif., 1985), Contemporary Mathematics 65, American Mathematical Society, Providence, RI, 1987, pp. 157--167; DOI 10.1090/conm/065/891246; Zbl 0635.03042. The volume is a conference proceedings, and no evidence that its papers were refereed was found, so refereed is not listed. The site's commentary does not cite the paper, so no curator credit exists and reviewed is not listed. The paper is paywalled and not held; its statement is taken from the zbMATH review and from Komjáth's restatement, and its proof is not checked. The volume carries only the year, so this page is dated the first of January 1987.

Attribution of $k=1$. Komjáth attributes the case $k=1$, in the form $\omega_1^2\to(\omega_1\alpha,3)^2$ for every $\alpha<\omega_1$, to Erdős and Hajnal, Some results and problems on certain polarized partitions, Acta Math. Acad. Sci. Hungar. 21 (1970), 369--392. That paper says in its §1 that it does not investigate relations for order types and gives its definitions for cardinals only, so the attribution is not borne out there. The ordinal relation $\omega_1^2\to(\mu,3)^2$ for $\mu<\omega_1^2$ is in Erdős and Hajnal, Ordinary partition relations for ordinal numbers, Period. Math. Hungar. 1 (1971), 171--185, which the zbMATH review (Zbl 0257.04004) states under GCH; whether that paper proves $k=1$ in ZFC is not established, so this page covers $k=1$ through the 1987 relation.

Depends on. Baumgartner and Hajnal 1987, positive relation.