Some Remarks on the Partition Calculus of Ordinals

摘要: One of the early partition relation theorems which include ordinals was observation Erdos and Rado [7] that if κ = cf(κ) > ω then Dushnik–Miller theorem can be sharpened to κ→(κ, + 1) 2 . The question on possible further extension this result answered by Hajnal who in [8] proved continuum hypothesis implies 1 ↛ (ω , 2) He actually stronger (ω: 2)) consistency κ↛(κ, later extensively studied. Baumgartner [1] it for every is successor a regular cardinal. Laver [9] showed Mahlo there forcing notion adds witness κ↛ (κ, preserves Mahloness, ω-Mahloness κ, etc. We notice connection with these results λ→(λ, holds λ singular, fact (μ: n )) In [11] Todorcevic cf(λ) ccc add counter-example give an alternative proof (Theorem 5) extend larger cardinals: GCH holds, cf (λ) (κ) -c.c. 6). remarked their problem paper [5] Galvin had →(ω he also asked 3) true. show Theorem negative consistent.