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Claim. The paper's main theorem, as its zbMATH review states it: if there is no scale of order type $\omega_1$ in $\omega^\omega$ under eventual domination, then $\omega_1^{\omega+2}\to(\omega_1^{\omega+2},\text{infinite path})^2$: every graph on $\omega_1^{\omega+2}$ has an infinite path or an independent set of order type $\omega_1^{\omega+2}$, the property asked by Problem 601 at the ordinal Erdős priced at $250. The review notes that the hypothesis follows from Martin's axiom and that $\omega_1^{\omega+2}$ is the first ordinal for which the property is not provable in ZFC.

Hypothesis. No scale of type $\omega_1$: no $\omega_1$-sequence of functions in $\omega^\omega$ is both increasing and cofinal under eventual domination. The hypothesis holds under Martin's axiom with the continuum above $\aleph_1$ and fails under the continuum hypothesis, where the diamond example shows that the conclusion can fail. The claim decides no instance in ZFC; Larson's later theorem under Martin's axiom extends the positive conclusion to every limit ordinal below the continuum.

Source. Jean A. Larson, A consequence of no short scale for ordinal graphs with no infinite paths, Journal of the London Mathematical Society (2) 33 (1986), no. 2, 193–202, doi:10.1112/jlms/s2-33.2.193; Zbl 0566.03030. The issue is dated April 1986 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 Journal of the London Mathematical Society. The site's commentary does not mention the paper and labels the problem OPEN, so reviewed is not listed.