Claims

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1971_01_01_hajnal: Hajnal (Proc. Nat. Acad. Sci. U.S.A., 1971) proves under the continuum hypothesis that omega_1^2 does not arrow (omega_1^2, 3)^2, so the relation Problem 1169 asks about cannot be refuted in ZFC; its provability is open.

1975_12_01_baumgartner: Baumgartner (Israel J. Math., 1975) proves that kappa^2 -> (kappa^2, 3)^2 for regular kappa implies the kappa-Souslin hypothesis, so a Suslin tree gives omega_1^2 -/-> (omega_1^2, 3)^2 and ZFC does not refute the relation.

1987_03_01_takahashi: Takahashi (Period. Math. Hungar., 1987) derives omega_1 omega_1 -/-> (omega_1 omega_1, 3)^2 from the stick principle, which CH implies, so ZFC does not refute the relation of Problem 1169.

1998_01_01_larson: Larson (Kluwer proceedings, 1998) derives omega_1^2 -/-> (omega_1^2, 3)^2 from the dominating number being aleph_1; claimed, since the volume is not shown to be refereed.

2026_08_13_golshani: An arXiv preprint (2026) gives a model of MA_{omega_1}(sigma-centered) with 2^{aleph_0} = aleph_2 in which omega_1^2 -/-> (omega_1^2, 3)^2; claimed.

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