Notre Dame Journal of Formal Logic

Stable Ramsey's Theorem and Measure

Damir D. Dzhafarov

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 WKL0 in the context of reverse mathematics.

Article information

Source
Notre Dame J. Formal Logic, Volume 52, Number 1 (2011), 95-112.

Dates
First available in Project Euclid: 13 December 2010

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1292249613

Digital Object Identifier
doi:10.1215/00294527-2010-039

Mathematical Reviews number (MathSciNet)
MR2747165

Zentralblatt MATH identifier
1217.03019

Subjects
Primary: 03D80: Applications of computability and recursion theory 05D10: Ramsey theory [See also 05C55] 03D32: Algorithmic randomness and dimension [See also 68Q30] 03F35: Second- and higher-order arithmetic and fragments [See also 03B30]

Keywords
Ramsey's theorem effective measure theory reverse mathematics

Citation

Dzhafarov, Damir D. Stable Ramsey's Theorem and Measure. Notre Dame J. Formal Logic 52 (2011), no. 1, 95--112. doi:10.1215/00294527-2010-039. https://projecteuclid.org/euclid.ndjfl/1292249613


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