Completely separable families and Ramsey's theorem

Abstract

Ramsey's theorem states that for every partition of the n-element subsets of ω (the nonnegative integers) into two classes, there exists an infinite set F ⊆ ω with all its n-element subsets in the same class. We show that if F is a completely separable family of infinite subsets of ω, then for every such partition there exists an infinite set F as above, with the additional property that F is a member of the family F . This answers a question of Hechler. We also strengthen our results along the lines of known generalizations of Ramsey's theorem and give a short proof of a theorem of Mathias.