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Claim. For every regular cardinal $\kappa$, if $\kappa^2\to(\kappa^2,3)^2$ then the $\kappa$-Souslin hypothesis holds, that is, there is no $\kappa$-Suslin tree (the paper's abstract, in the corpus's words). At $\kappa=\omega_1$, a Suslin tree therefore gives $\omega_1^2\not\to(\omega_1^2,3)^2$, the relation of Problem 1169. A Suslin tree exists in some model of ZFC, for example in Gödel's constructible universe by Jensen's theorem, so ZFC does not refute the relation. Suslin trees exist in models where CH fails, so this route does not need CH.

Covers. One side of an independence result: the relation holds in every model with a Suslin tree, so ZFC does not refute it, but nothing here shows that ZFC does not prove it. One side alone leaves the question open, so the result leaves Problem 1169 open. It would be settled as independent by a model of $\omega_1^2\to(\omega_1^2,3)^2$, which no source records, and as proved by a proof of the negative relation in ZFC alone.

Depends on. No page of this wiki; the step to the relation's consistency uses Jensen's theorem that a Suslin tree exists in $L$.

Source. J. E. Baumgartner, Partition relations for uncountable ordinals, Israel J. Math. 21 (1975), no. 4, 296-307, doi:10.1007/BF02757991. The issue is dated December 1975 and the record carries no day, so this page's date is the first of that month. Komjáth's Problem 13 commentary (source card) records the Suslin-tree extension of Hajnal's theorem.

Acceptance. Refereed: the result is a journal paper in the Israel Journal of Mathematics.