We provide a much shorter proof of the following partition theorem of P. Erd˝ os and R. Rado: If X is an uncountable linear order into which neither ω 1 nor ω * 1 embeds, then X → (α, 4) 3 for every ordinal α < ω + ω. We also provide two counterexamples to possible generalizations of this theorem, one of which answers a question of E. C. Milner and K.… (More)
Consider an arbitrary partition of the triples of all countable or-dinals 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 n-element set. That is, we prove that ω 1 → (ω + m,… (More)
We prove that if P is a partial order and P → (ω) 1 ω , then (a) P → (ω + ω + 1, 4) 3 , and (b) P → (ω + m, n) 3 for each m, n < ω. Together these results represent the best progress known to us on the following question of P. Erd˝ os and others. If P → (ω) 1 ω , then does P → (α, n) 3 for each α < ω 1 and each n < ω?