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Claim. Theorem 5 of the paper (printed p. 181, in §4, "Some positive relations") states that for an arbitrary ordinal $\xi$, every $k<\omega$ and every $t$ with $1\le t<\omega$,

$$\omega_{\xi+1}^{(t+1)(k+1)}\to(\mu,t+2)^2 \quad\text{for every }\mu<\omega_{\xi+1}^{k+2}.$$

The results of the paper that use the generalized continuum hypothesis open with "Assume G.C.H.", Theorem 4 and Corollary 3 among them; Theorem 5 carries no hypothesis beyond ZFC. With $\xi=0$, the paper's $k=0$, $t=1$ and $\mu=\omega_1\omega<\omega_1^2$ it gives

$$\omega_1^2\to(\omega_1\omega,3)^2,$$

the case $k=1$ of Problem 1171: every coloring of the pairs from $\omega_1^2$ with two colors has a set of order type $\omega_1\omega$ homogeneous in color $0$ or a triangle homogeneous in color $1$. The same theorem is recorded for the triangle case of Problem 597 on the E0597 claim page.

Covers. The instance $k=1$ of the question, in ZFC. It does not address $k\ge2$: the case $k=2$, which contains $k=1$, is on the accepted page Baumgartner and Hajnal 1987, and the question is open in ZFC for $k\ge3$.

Source. P. Erdős and A. Hajnal, Ordinary partition relations for ordinal numbers, Periodica Mathematica Hungarica 1 (1971), no. 3, 171–185, doi:10.1007/BF02029142; the issue is dated September 1971 and carries no day, so this page is dated the first of that month. The second paper link is the open scan in the Rényi Institute's Erdős archive, on which the statement was read on the page image and the paper's uses of G.C.H. in the text layer. The zbMATH review (Zbl 0257.04004) summarizes the paper's relations, this one among them, under GCH, which the paper itself assumes only where it says so. The proof, by induction on $t$, begins on p. 182 with the case $t=1$ and rests on a lemma derived from the free-set theorem of Erdős, Hajnal and Milner; it was not checked, and nothing on this page is independently reviewed by this project.

Acceptance. Refereed: a journal paper in Periodica Mathematica Hungarica. The erdosproblems.com commentary on Problem 1171 does not cite the paper, so reviewed is not listed.