A recursion theoretic analysis of the clopen Ramsey theorem

Journal of Symbolic Logic 49 (2):376-400 (1984)
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Solovay has shown that if F: [ω] ω → 2 is a clopen partition with recursive code, then there is an infinite homogeneous hyperarithmetic set for the partition (a basis result). Simpson has shown that for every 0 α , where α is a recursive ordinal, there is a clopen partition F: [ω] ω → 2 such that every infinite homogeneous set is Turing above 0 α (an anti-basis result). Here we refine these results, by associating the "order type" of a clopen set with the Turing complexity of the infinite homogeneous sets. We also consider the Nash-Williams barrier theorem and its relation to the clopen Ramsey theorem



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Partial impredicativity in reverse mathematics.Henry Towsner - 2013 - Journal of Symbolic Logic 78 (2):459-488.

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Recursive well-founded orderings.Keh-Hsun Chen - 1978 - Annals of Mathematical Logic 13 (2):117-147.
Recursice well-founded orderings.D. -H. Chen - 1978 - Annals of Mathematical Logic 13 (2):117.

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