More on Partitioning Triples of Countable Ordinals

Consider an arbitrary partition of the triples of all countable ordinals into two classes. We show that either for each finite ordinal m the first class of the partition contains all triples from a set of type ω +m, or for each finite ordinal n the second class of the partition contains all triples of an nelement set. That is, we prove that ω1 → (ω + m,n)3 for each pair of finite ordinals m and n. 1. Background and motivation If A is a set of ordinals and β is an ordinal, then [A] denotes the set of all subsets of A which are order isomorphic to β. If α, ν, and {βi | i < ν} are ordinals and m < ω, then the ordinary Ramsey relation α → (βi)i<ν means that for each partition f : [α] → ν there are i < ν and X ∈ [α]i with f“[X] = {i}. As usual, if ν < ω, then this relation might be written α → (β0, . . . , βν−1); if βi = β for all i < ν, then it might be written α → (β)ν ; and the negation of any such relation is indicated by replacing the → with . The study of these relations (as well as all of the notation defined above) was introduced by P. Erdős and R. Rado in [2]. The ordinary Ramsey theory of the countable ordinals is the theory of such partition relations with α ≤ ω1, those relations which describe the ordinary Ramseytheoretic properties of individual countable ordinals or of the totality of all countable ordinals. This theory has been studied thoroughly and quite successfully. Our understanding of it, as witnessed by the results below, is almost complete. In each of the relations listed below, α is an arbitrary countable ordinal, and m and n are arbitrary finite ordinals, unless otherwise indicated. (1) ω → (ω)n (F. P. Ramsey in [10]). (2) (a) α (ω + 1, ω) (P. Erdős and R. Rado in [2]). (b) If m ≥ 3, then α (ω + 1,m+ 1) (P. Erdős and R. Rado in [2]). (3) (a) ω1 → (ω1)ω. (b) If m ≥ 2, then ω1 (m+ 1)ω (P. Erdős and R. Rado in [2]). (4) ω1 → (α)n (J. Baumgartner and A. Hajnal in [1]). (5) (a) ω1 (ω1)2 (W. Sierpiński in [11]). Received by the editors March 1, 2005 and, in revised form, October 25, 2005. 2000 Mathematics Subject Classification. Primary 03E05, 04A20; Secondary 05A18, 05D10.