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# Ramsey's theorem and recursion theory

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Let N be the set of natural numbers. If AN, let [A]n denote the class of all n-element subsets of A. If P is a partition of [N]n into finitely many classes C1, …, Cp, let H(P) denote the class of those infinite sets AN such that [A]nCi for some i. Ramsey's theorem [8, Theorem A] asserts that H(P) is nonempty for any such partition P. Our purpose here is to study what can be said about H(P) when P is recursive, i.e. each Ci, is recursive under a suitable coding of [N]n. We show that if P is such a recursive partition of [N]n, then H(P) contains a set which is Πn0 in the arithmetical hierarchy. In the other direction we prove that for each n ≥ 2 there is a recursive partition P of [N]n into two classes such that H(P) contains no Σn0 set. These results answer a question raised by Specker [12].

A basic partition is a partition of [N]2 into two classes. In §§2, 3, and 4 we concentrate on basic partitions and in so doing prepare the way for the general results mentioned above. These are proved in §5. Our “positive” results are obtained by effectivizing proofs of Ramsey's theorem which differ from the original proof in [8]. We present these proofs (of which one is a generalization of the other) in §§4 and 5 in order to clarify the motivation of the effective versions.

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[1]Friedberg, R. M., Two recursively enumerable sets of incomparable degree of unsolvability (solution of Post's problem 1944), Proceedings of the National Academy of Sciences of the United States of America, vol. 43 (1957), pp. 236238.
[2]Jockusch, C. G., Semirecursive sets and positive reducibility, Transactions of the American Mathematical Society, vol. 131 (1968), pp. 420436.
[3]Jockusch, C. G., The degrees of bi-immune sets, Zeitschrift für mathematische Logik and Grundlagen der Mathematik, vol. 15 (1969), pp. 135140.
[4]Jockusch, C. G., Ramsey's theorem and recursion theory, Notices of the American Mathematical Society, vol. 17 (1970), pp. 672673.
[5]Jockusch, C. G. and McLaughlin, T. G., Countable retracing functions and Π20 predicates, Pacific Journal of Mathematics, vol. 30 (1969), pp. 6793.
[6]Jockusch, C. G. and Soare, R. I., Π20 classes and degrees of theories (to appear).
[7]McLaughlin, T. G., Splitting and decomposition by regressive sets. II, Canadian Journal of Mathematics, vol. 19 (1967), pp. 291311.
[8]Ramsey, F. P., On a problem in formal logic, Proceedings of the London Mathematical Society, vol. 30 (1930), pp. 264286.
[9]Rogers, H., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.
[10]Sacks, G. E., A minimal degree less than O', Bulletin of the American Mathematical Society, vol. 67 (1961), pp. 416419.
[11]Shoenfield, J. R., On degrees of unsolvability, Annals of Mathematics, series 2, vol. 69 (1959), pp. 644653.
[12]Specker, E., Ramsey's theorem does not hold in recursive set theory, Studies in logic and the foundations of mathematics, North-Holland, Amsterdam, 1971.
[13]Yates, C. E. M., Arithmetical sets and retracing functions, Zeitschrift für mathematische Logik and Grundlagen der Mathematik, vol. 13 (1967), pp. 193204.
[14]Yates, C. E. M., A note on arithmetical sets of indiscernibles, Studies in logic and the foundations of mathematics, North-Holland, Amsterdam, 1971, pp. 443451.
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The Journal of Symbolic Logic
• ISSN: 0022-4812
• EISSN: 1943-5886
• URL: /core/journals/journal-of-symbolic-logic
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