---

Claim. The paper's main theorem, as its zbMATH review states it: under Martin's axiom, $\alpha\to(\alpha,\text{infinite path})^2$ for every limit ordinal $\alpha<2^{\aleph_0}$, that is, every graph on such an $\alpha$ has an infinite path or an independent set of order type $\alpha$, the property asked by Problem 601. The paper also proves a reduction rule the review records: if $\alpha=\alpha_0+\cdots+\alpha_{n-1}$ with $\alpha>\alpha_0\ge\cdots\ge\alpha_{n-1}$ and no $\alpha_i$ the sum of two smaller ordinals, then $\alpha$ has the property if and only if every $\alpha_i$ has it.

Hypothesis. Martin's axiom. Under MA with $2^{\aleph_0}=\aleph_2$, which is consistent relative to ZFC (Solovay and Tennenbaum, Ann. of Math. 94, 1971), the theorem covers every limit $\alpha<\omega_2$, in particular $\alpha=\omega_1^{\omega+2}$, the case Erdős priced at $250; under the continuum hypothesis it covers only the countable limit ordinals, where the property is a theorem of ZFC by Erdős, Hajnal and Milner. The claim decides no instance in ZFC. Set against the diamond example of the same issue, it makes the case $\alpha=\omega_1^{\omega+2}$ independent of ZFC; the problem page records the argument.

Source. Jean A. Larson, Martin's axiom and ordinal graphs: large independent sets or infinite paths, Annals of Pure and Applied Logic 47 (1990), no. 1, 31–39, doi:10.1016/0168-0072(90)90015-T; Zbl 0703.03029; the site's key [La90]. 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 credits Larson with the result under Martin's axiom, but the site labels the problem OPEN, so the curator's credit is not acceptance of a claim and reviewed is not listed.