Claims

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1987_01_01_baumgartner_hajnal: Baumgartner and Hajnal (Contemp. Math. 65, 1987) prove that (kappa^+)^2 -> (kappa^+ kappa, 3, 3)^2 for regular kappa with kappa^(<kappa) = kappa; for kappa = omega this is the ZFC case k = 2 of Problem 1171, containing k = 1.

1989_01_01_baumgartner: Baumgartner (Lecture Notes in Math. 1401, 1989) proved that Martin's axiom for aleph_1 dense sets makes omega_1 omega a partition ordinal; the catalog relation follows for every finite k, so it holds in a model of ZFC.

2026_09_05_gao: An unrefereed, AI-assisted Zenodo deposit proving the exact relation for every finite k from Martin's axiom for aleph_1 dense sets by a color-reduction lemma; filed on the site as a partial claim, then withdrawn by its author.

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