Claims

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1956_12_01_specker: Specker (Comment. Math. Helv. 31, 1956/57) proves omega^2 -> (omega^2, n)^2 for every finite n and omega^m -/-> (omega^m, 3)^2 for finite m >= 3: yes at beta = 2 and no at every finite beta >= 3; refereed.

1972_05_01_chang: Chang (J. Combin. Theory Ser. A 12, 1972) proves omega^omega -> (omega^omega, 3)^2, the case beta = omega of the question, yes; refereed.

1975_01_01_galvin_larson: Galvin and Larson (Fund. Math. 82, 1974/75) show that a countable partition ordinal is 0, 1, omega^2 or omega^(omega^gamma), so omega^beta fails the relation for every decomposable beta >= 3; refereed.

2010_05_13_schipperus: Schipperus (Ann. Pure Appl. Logic 161, 2010) proves the relation for beta = omega^gamma with gamma the sum of one or two indecomposables and refutes it for four or more; three summands stay open; refereed.

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