Ramsey's theorem and cone avoidance
- 1 June 2009
- journal article
- Published by Cambridge University Press (CUP) in The Journal of Symbolic Logic
- Vol. 74 (2), 557-578
- https://doi.org/10.2178/jsl/1243948327
Abstract
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 ≰rH, 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.Keywords
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