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Claim. The stick principle says that there are $\aleph_1$ countable subsets of $\omega_1$ such that every uncountable subset of $\omega_1$ contains one of them. Takahashi proves that the stick principle implies $\omega_1\cdot\omega_1\not\to(\omega_1\cdot\omega_1,3)^2$, as the zbMATH review (Zbl 0639.03053) gives the result. Since $\omega_1\cdot\omega_1$ is the ordinal $\omega_1^2$, this is the relation of Problem 1169. CH implies the stick principle, and the principle 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.

Covers. One side of an independence result: the relation holds in every model of the stick principle, 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.

Source. J. Takahashi, Two negative partition relations, Period. Math. Hungar. 18 (1987), no. 1, 1-6, doi:10.1007/BF01849028. The issue is dated March 1987 and the record carries no day, so this page's date is the first of that month.

Acceptance. Refereed: the result is a journal paper in Periodica Mathematica Hungarica.