Some remarks on the partition calculus of ordinals

Extract

One of the early partition relation theorems which include ordinals was the observation of Erdös and Rado [7] that if κ = cf(κ) > ω then the Dushnik–Miller theorem can be sharpened to κ→(κ, ω + 1)². The question on the possible further extension of this result was answered by Hajnal who in [8] proved that the continuum hypothesis implies ω₁ ↛ (ω₁, ω + 2)². He actually proved the stronger result ω₁ ↛ (ω: 2))². The consistency of the relation κ↛(κ, (ω: 2))² was later extensively studied. Baumgartner [1] proved it for every κ which is the successor of a regular cardinal. Laver [9] showed that if κ is Mahlo there is a forcing notion which adds a witness for κ↛ (κ, (ω: 2))² and preserves Mahloness, ω-Mahloness of κ, etc. We notice in connection with these results that λ→(λ, (ω: 2))² holds if λ is singular, in fact λ→(λ, (μ: n))² for n < ω, μ < λ (Theorem 4).

In [11] Todorčević proved that if cf(λ) > ω then a ccc forcing can add a counter-example to λ→(λ, ω + 2)². We give an alternative proof of this (Theorem 5) and extend it to larger cardinals: if GCH holds, cf (λ) > κ = cf (κ) then < κ-closed, κ⁺-c.c. forcing adds a counter-example to λ→(λ, κ + 2)² (Theorem 6).

Erdös and Hajnal remarked in their problem paper [5] that Galvin had proved ω₂→(ω₁, ω + 2)² and he had also asked if ω₂→(ω₁, ω + 3)² is true. We show in Theorem 1 that the negative relation is consistent.