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Ramsey's theorem and cone avoidance

Published online by Cambridge University Press:  12 March 2014

Damir D. Dzhafarov and
Carl G. Jockusch Jr.
Damir D. Dzhafarov
Affiliation:
Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, Il 60637, E-mail: damir@math.uchicago.edu
Carl G. Jockusch Jr.
Affiliation:
Department of Mathematics, University of Illinoisat Urbana-Champaign, 1409 W. Green St., Urbana. Il 61801, E-mail: jockusch@math.uiuc.edu
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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.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 2 , June 2009, pp. 557 - 578
Copyright
Copyright © Association for Symbolic Logic 2009

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References

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