Ramsey’s theorem and cone avoidance. (English) Zbl 1166.03021

Summary: It was shown by P. A. Cholak, C. G. Jockusch, and T. A. Slaman [J. Symb. Log. 66, No. 1, 1–55 (2001; Zbl 0977.03033)] that every computable 2-coloring of pairs admits an infinite low\(_{2}\) homogeneous set \(H\). We answer a question of the same authors by showing that \(H\) may be chosen to satisfy in addition \(C \not \leq _{T} H\), 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 low\(_{2}\) infinite homogeneous sets whose degrees form a minimal pair.


03D80 Applications of computability and recursion theory
03B30 Foundations of classical theories (including reverse mathematics)
03D30 Other degrees and reducibilities in computability and recursion theory
05D10 Ramsey theory


Zbl 0977.03033
Full Text: DOI Link


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