Journal of Symbolic Logic

Ramsey's theorem and cone avoidance

Damir D. Dzhafarov and Carl G Jockusch, Jr.

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It was shown by Cholak, Jockusch, and Slaman that every computable 2-coloring of pairs admits an infinite low2 homogeneous set H. We answer a question of the same authors by showing that H may be chosen to satisfy in addition C ≰T H, where C is a given noncomputable set. This is shown by analyzing a new and simplified proof of Seetapun's cone avoidance theorem for Ramsey's theorem. We then extend the result to show that every computable 2-coloring of pairs admits a pair of low2 infinite homogeneous sets whose degrees form a minimal pair.

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J. Symbolic Logic, Volume 74, Issue 2 (2009), 557-578.

First available in Project Euclid: 2 June 2009

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Dzhafarov, Damir D.; Jockusch, Jr., Carl G. Ramsey's theorem and cone avoidance. J. Symbolic Logic 74 (2009), no. 2, 557--578. doi:10.2178/jsl/1243948327.

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