## Journal of Symbolic Logic

### Ramsey's theorem and cone avoidance

#### 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 ≰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.

#### Article information

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

Dates
First available in Project Euclid: 2 June 2009

https://projecteuclid.org/euclid.jsl/1243948327

Digital Object Identifier
doi:10.2178/jsl/1243948327

Mathematical Reviews number (MathSciNet)
MR2518811

Zentralblatt MATH identifier
1166.03021

#### Citation

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. https://projecteuclid.org/euclid.jsl/1243948327