Some Ramsey-type theorems for countably determined sets

Let X be an infinite internal set in an w1-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 o...

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Bibliographic Details
Published in:Archive for mathematical logic 2002-10, Vol.41 (7), p.619-630
Main Authors: Mlcek, Josef, Zlatos, Pavol
Format: Article
Language:English
Subjects:
Theorems
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Summary:Let X be an infinite internal set in an w1-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 [subset of or = to] 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 on[subset of or = to] [[X]k]m, with arbitrary standard k greater than or equal to 1, m greater than or equal to 2. In the course of the proof certain minimal condensating countably determined sets will be described.
ISSN:0933-5846
1432-0665
DOI:10.1007/s001530100129