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Claim. The paper's theorem, as its zbMATH review states it: assume the generalized continuum hypothesis; then for every integer $n\ge2$ there is a cofinal set of ordinals $\alpha<\omega_n$ with $\alpha\not\to(\alpha,\text{infinite path})^2$, that is, a graph on $\alpha$ with no infinite path and no independent set of order type $\alpha$. Under GCH, then, the property asked by Problem 601 fails for cofinally many limit ordinals below every $\omega_n$ with $n\ge2$, so in a model of GCH the answer to the general question is not every limit ordinal.

Hypothesis. The generalized continuum hypothesis, which holds in $L$. The claim decides no instance in ZFC. The ordinals the theorem produces are not identified here; below $\omega_1^{\omega+2}$ every limit ordinal has the property in ZFC by Erdős, Hajnal and Milner, so the failing ordinals below $\omega_2$ lie at or above $\omega_1^{\omega+2}$, where the diamond example later made every limit ordinal fail.

Source. Jean A. Larson, A GCH example of an ordinal graph with no infinite path, Transactions of the American Mathematical Society 303 (1987), no. 1, 383–393, doi:10.1090/S0002-9947-1987-0896028-6; Zbl 0638.05002. The issue is dated September 1987 and carries no day, so this page is dated the first of that month. The paper is 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 Transactions of the American Mathematical Society. The site's commentary does not mention the paper and labels the problem OPEN, so reviewed is not listed.