Ramsey's theorem for computably enumerable colorings

Journal of Symbolic Logic 66 (2):873-880 (2001)
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Abstract

It is shown that for each computably enumerable set P of n-element subsets of ω there is an infinite Π 0 n set $A \subseteq \omega$ such that either all n-element subsets of A are in P or no n-element subsets of A are in P. An analogous result is obtained with the requirement that A be Π 0 n replaced by the requirement that the jump of A be computable from 0 (n) . These results are best possible in various senses

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Theory of Recursive Functions and Effective Computability.Hartley Rogers - 1971 - Journal of Symbolic Logic 36 (1):141-146.
On the Strength of Ramsey's Theorem.David Seetapun & Theodore A. Slaman - 1995 - Notre Dame Journal of Formal Logic 36 (4):570-582.
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The Degrees of Hyperimmune Sets.Webb Miller & D. A. Martin - 1968 - Mathematical Logic Quarterly 14 (7‐12):159-166.
A cohesive set which is not high.Carl Jockusch & Frank Stephan - 1993 - Mathematical Logic Quarterly 39 (1):515-530.

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