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Ramsey's theorem and cone avoidance

  • Damir D. Dzhafarov (a1) and Carl G. Jockusch (a2)

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 CrH, 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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References

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[1]Cholak, Peter A., Jockusch, Carl G. Jr., and Slaman, Theodore A., On the strength of Ramsey's theorem for pairs, this Journal, vol. 66 (2001), no. 1, pp. 155.
[2]Hirschfeldt, Denis R., Jockusch, Carl G. Jr., Kjos-Hanssen, Bjørn, Lempp, Steffen, and Slaman, Theodore A., The strength of some combinatorial principles related to Ramsey's theorem for pairs, Computational Prospects of Infinity, Part II: Presented Talks, Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore (Chong, C.T., Feng, Q., Slaman, T., Woodin, H., and Yang, Y., editors), vol. 15, World Scientific, New Jersey, London, Singapore, Beijing, Shanghai, Hong Kong, Taipei, Chennai, 2008, pp. 143161.
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[11]Simpson, Stephen G., Subsystems of second order arithmetic, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1999.
[12]Soare, Robert I., Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1987, A study of computable functions and computably generated sets.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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