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Claim. The paper's theorem, as its zbMATH review states it: if Jensen's principle $\diamondsuit$ holds, then $\alpha\not\to(\omega_1^{\omega+2},\text{infinite path})^2$ for every $\alpha<\omega_2$: there is a graph on $\alpha$ with no infinite path and no independent set of order type $\omega_1^{\omega+2}$. For a limit ordinal $\alpha$ with $\omega_1^{\omega+2}\le\alpha<\omega_2$ an independent set of type $\alpha$ would contain one of type $\omega_1^{\omega+2}$, so under $\diamondsuit$ no such $\alpha$ has the property asked by Problem 601; in particular $\alpha=\omega_1^{\omega+2}$, the case Erdős offered a prize for, fails. Erdős reports the result under Problem 10 of Erdős 1987 (printed p. 226) as recent work of Larson and Baumgartner, then to appear: it is consistent that every $\alpha<\omega_2$ fails to have an independent set of type $\omega_1^{\omega+2}$ or an infinite path.

Hypothesis. Jensen's $\diamondsuit$, which holds in $L$ and implies the continuum hypothesis, so the model has $\omega_1^{\omega+2}<\omega_2=(2^{\aleph_0})^+$. The claim decides no instance in ZFC. Set against Larson's theorem under Martin's axiom in the same issue, which gives the property to every limit $\alpha<2^{\aleph_0}$, it makes the case $\alpha=\omega_1^{\omega+2}$ independent of ZFC; the problem page records the argument.

Source. James E. Baumgartner and Jean A. Larson, A diamond example of an ordinal graph with no infinite paths, Annals of Pure and Applied Logic 47 (1990), no. 1, 1–10, doi:10.1016/0168-0072(90)90013-R; Zbl 0703.03028. The issue is dated April 1990 and carries no day, so this page is dated the first of that month. The paper is paywalled and not held; its statement is taken from the zbMATH review. Nothing on this page is independently reviewed by this project.

Acceptance. Refereed: a journal paper in the Annals of Pure and Applied Logic. The site's commentary does not mention the paper and labels the problem OPEN, so reviewed is not listed.