Some Ramsey-type theorems for countably determined sets

Abstract.

 Let X be an infinite internal set in an ω¹-saturated nonstandard universe. Then for any coloring of [X]^k, such that the equivalence E of having the same color is countably determined and there is no infinite internal subset of [X]^k with all its elements of different colors (i.e., E is condensating on X), there exists an infinite internal set Z⊆X such that all the sets in [Z]^k have the same color. This Ramsey-type result is obtained as a consequence of a more general one, asserting the existence of infinite internal Q-homogeneous sets for certain Q ⊆ [[X] ^k ] ^m, with arbitrary standard k≥ 1, m≥ 2. In the course of the proof certain minimal condensating countably determined sets will be described.