Stable Ramsey's Theorem and Measure

Notre Dame Journal of Formal Logic 52 (1):95-112 (2011)
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Abstract

The stable Ramsey's theorem for pairs has been the subject of numerous investigations in mathematical logic. We introduce a weaker form of it by restricting from the class of all stable colorings to subclasses of it that are nonnull in a certain effective measure-theoretic sense. We show that the sets that can compute infinite homogeneous sets for nonnull many computable stable colorings and the sets that can compute infinite homogeneous sets for all computable stable colorings agree below $\emptyset'$ but not in general. We also answer the analogs of two well-known questions about the stable Ramsey's theorem by showing that our weaker principle does not imply COH or WKL 0 in the context of reverse mathematics

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Citations of this work

Reverse mathematics and a Ramsey-type König's Lemma.Stephen Flood - 2012 - Journal of Symbolic Logic 77 (4):1272-1280.
A strong law of computationally weak subsets.Bjørn Kjos-Hanssen - 2011 - Journal of Mathematical Logic 11 (1):1-10.
Reverse Mathematics and Ramsey Properties of Partial Orderings.Jared Corduan & Marcia Groszek - 2016 - Notre Dame Journal of Formal Logic 57 (1):1-25.

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References found in this work

Subsystems of Second Order Arithmetic.Stephen G. Simpson - 1999 - Studia Logica 77 (1):129-129.
Ramsey's theorem and recursion theory.Carl G. Jockusch - 1972 - Journal of Symbolic Logic 37 (2):268-280.
Measure theory and weak König's lemma.Xiaokang Yu & Stephen G. Simpson - 1990 - Archive for Mathematical Logic 30 (3):171-180.
A cohesive set which is not high.Carl Jockusch & Frank Stephan - 1993 - Mathematical Logic Quarterly 39 (1):515-530.

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