A short proof of a partition theorem for the ordinal ωω

Countable partition ordinals

“…Introduce a game between two players, called the Builder and the Architect, where the Builder constructs a pair of trees (S, T ) and the Architect provides information which restricts some of the Builder's moves. 4. Then prove a Ramsey dichotomy for that game: that for some infinite H ⊆ ω if the Builder plays sufficiently large numbers inside H then either the Architect has a winning strategy, or every strategy for the Builder is a winning strategy.…”

“…Let h be the next sufficiently large element in H. Define ∆ x = {h}. 4. α x is a successor, x ∈ G(T ) and x ∈T .…”

Open problems of Paul Erd�s in graph theory

“…Chang [43] proved ω ω → (ω ω , 3) 2 . Milner [187] generalized the proof of Chang to show ω ω → (ω ω β , n) 2 for n < ω, and Larson [173] gave simpler proof.…”

Negative Partition Relations for Ordinals ωωα

“…Besides |, only two countable ordinals are known to belong to the first group, called partition ordinals, namely | 2 (Specker [10]) and | | (Chang [2], Milner, unpublished, and Larson [7]). The second group (reluctantly termed anti-partition ordinals by Larson) contains additively decomposable ordinals, i.e., ordinals that can be written as the sum of two smaller ordinals, | 3 [10], and ordinals which can be pinned to | 3 .…”