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Claim. Theorem 7 of the paper (§6, printed p. 358): let $S$ be an ordered set of type $\omega\theta<\omega_1^{\omega+2}$ and let $G=(S,E)$ be any graph on $S$ with no infinite path; then $S$ has an independent set $S'$ of the same type $\omega\theta$. Every limit ordinal is of the form $\omega\theta$, so every limit $\alpha<\omega_1^{\omega+2}$ has the property asked by Problem 601: a graph on $\alpha$ has an infinite path or an independent set of order type $\alpha$. The proof uses the paper's Theorem 5 on set mappings, and the remark after the theorem says that this is what restricts the type below $\omega_1^{\omega+2}$: Theorem 5 fails for larger types of cardinality $\aleph_1$, while the authors suspect that Theorem 7 holds for arbitrary $\theta$ and cannot prove it. Erdős records the result under Problem 10 of Erdős 1987 (printed p. 226).

Covers. Every limit ordinal $\alpha<\omega_1^{\omega+2}$, positively. It says nothing about $\alpha=\omega_1^{\omega+2}$, the first case the paper leaves open and the case Erdős later priced at $250, nor about any larger limit ordinal.

Source. P. Erdős, A. Hajnal and E. C. Milner, Set mappings and polarized partition relations, in Combinatorial theory and its applications, I (Proc. Colloq., Balatonfüred, 1969), Colloquia Mathematica Societatis János Bolyai 4, North-Holland, Amsterdam, 1970, pp. 327–363; Zbl 0215.329; the site's key [EHM70]. The paper link is the open scan in the Rényi Institute's Erdős archive; the source card records the paper. The volume carries only the year, so this page is dated the first of January 1970. Nothing on this page is independently reviewed by this project.

Acceptance. Reviewed: the theorem is restated as proved and built on in the refereed papers of Jean A. Larson (1986, 1987, 1990) and of Baumgartner and Larson (1990), whose zbMATH reviews repeat it: a documented acceptance by named experts independent of the authors. Not refereed: the colloquium proceedings carry no evidence that their papers were refereed. The site's commentary credits the authors with the result, but the site labels the problem OPEN, so the curator's credit is not acceptance of a claim.