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Claim. Let $2\le r<\omega$, let $\gamma\ge2$, and let $\kappa_\alpha>r$ for $\alpha<\gamma$. Section 24 of Erdős, Hajnal, Máté and Rado's monograph Combinatorial Set Theory: Partition Relations for Cardinals proves that $2^\lambda\to(\kappa_\alpha+1)^{r+1}_{\alpha<\gamma}$ implies $\lambda\to(\kappa_\alpha)^r_{\alpha<\gamma}$ in each of five cases: all $\kappa_\alpha$ are finite; $\kappa_0$ and $\kappa_1$ are infinite and $\kappa_0$ is regular; $r\ge3$ and $\kappa_0$ is infinite and regular; $r\ge3$ and $\kappa_0$ and $\kappa_1$ are infinite; $r\ge4$ and $\kappa_0$ is infinite. Komjáth's Problem 2 commentary (source card) records the same five cases as proved, citing that section of the monograph, and the site's remark lists them too.

Covers. The implication of Problem 1167 in the five cases above, under the list's conditions $2\le r<\omega$, $\gamma\ge2$ and $\kappa_\alpha>r$. Without the condition on $\gamma$ the third case would include the literal counterexample $\gamma=1$, $\kappa_0=\lambda^+$, recorded on Zeraoulia's page. The case Erdős and Hajnal named as the hardest, $r=2$ with one singular $\kappa_\alpha$ and the others finite, is not covered.

Depends on. No page of this wiki.

Source. P. Erdős, A. Hajnal, A. Máté and R. Rado, Combinatorial Set Theory: Partition Relations for Cardinals, Studies in Logic and the Foundations of Mathematics 106, North-Holland, Amsterdam, 1984 (MR 795592). The record carries no day or month, so this page's date is the first of the year.

Acceptance. None that the schema counts. The result is in a monograph, not a journal paper, so it is not refereed. The site labels the problem OPEN, so its remark crediting the cases is commentary, not acceptance. The claim is therefore claimed.