A dual form of Ramsey's Theorem

Abstract

Let k ϵ ω, where ϵ is the set of all natural numbers. Ramsey's Theorem deals with colorings of the k-element subsets of ω. Our dual form deals with colorings of the k-element partitions of ω. Let ( ω) k (respectively ( ω) ω ) be the set of all partitions of ω having exactly k (respectively infinitely many) blocks. Given X ϵ ( ω) ω let ( X) k be the set of all Y ϵ ( ω) k such that Y is coarser than X. Dual Ramsey Theorem. If ( ω) k = C 0 ∪ … ∪ C t−1 where each C i is Borel then there exists X ϵ ( ω) ω such that ( X) k ⊆ C i for some i < l. Dual Galvin-Prikry Theorem. Same as before with k replaced by ω. We also obtain dual forms of theorems of Ellentuck and Mathias. Our results also provide an infinitary generalization of the Graham-Rothschild “parameter set” theorem [ Trans. Amer. Math. Soc. 159 (1971), 257–292] and a new proof of the Halpern-Läuchli Theorem [ Trans. Amer. Math. Soc. 124 (1966), 360–367].