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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.

MSC:

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

Citations:

Zbl 0977.03033
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References:

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