A dual form of Ramsey's Theorem

Let k?ω, where ? 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? (ω)^ω let (X)^k be the set of all Y? (ω)^k such that Y is coarser than X. Dual Ramsey Theorem. If (ω)^k = C₀ ∪ … ∪ C_t?1 where each C_i is Borel then there exists X? (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].