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Claim. If the dominating number $\mathfrak d$ equals $\aleph_1$, that is, if some family of $\aleph_1$ functions from $\omega$ to $\omega$ eventually dominates every such function, then $\omega_1^2\not\to(\omega_1^2,3)^2$, as the zbMATH review (Zbl 0890.03018) gives the result. This is the relation of Problem 1169. CH implies $\mathfrak d=\aleph_1$, and $\mathfrak d=\aleph_1$ is consistent with the failure of CH, so the relation holds in models of ZFC with and without CH, and ZFC does not refute it. Chen, Garti and Weinert (arXiv:1801.00238, Fact 1.1) list this result beside Takahashi's.

Covers. One side of an independence result: the relation holds in every model of $\mathfrak d=\aleph_1$, 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, even once accepted, leaves Problem 1169 open; the accepted pages of Hajnal, Baumgartner and Takahashi already settle the same side.

Depends on. No page of this wiki.

Source. J. A. Larson, An ordinal partition from a scale, in Set Theory: Techniques and Applications (Curaçao 1995 and Barcelona 1996), Kluwer, Dordrecht, 1998, 109-125 (Zbl 0890.03018). The record carries no day or month, so this page's date is the first of the year.

Acceptance. None that the schema counts. The result is in a proceedings volume with no record that it was refereed, and the site does not credit it. The claim is therefore claimed.