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Ramsey's theorem for computably enumerable colorings

Published online by Cambridge University Press:  12 March 2014

Tamara J. Hummel  and
Carl G. Jockusch Jr.
Tamara J. Hummel
Affiliation:
Department of Mathematics, Allegheny College, 520 N. Main St., Meadville, PA 16335., USA, E-mail: thummel@allegheny.edu
Carl G. Jockusch Jr.
Affiliation:
Department of Mathematics, University of Illinois, 1409 W. Green St., Urbana, Il 61801, USA, E-mail: jockusch@math.uiuc.edu
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Abstract

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

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 2 , June 2001 , pp. 873 - 880
Copyright
Copyright © Association for Symbolic Logic 2001

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References

REFERENCES

[1]Cenzer, D., classes, Handbook of computability theory (Griffor, E., editor). Studies in Logic and the Foundations of Mathematics, vol. 140, Elsevier, 1999, pp. 37–85.CrossRefGoogle Scholar
[2]Cholak, P., Jockusch, C. G. Jr., and Slaman, T., On the strength of Ramsey's theorem for pairs, to appear in This Journal.Google Scholar
[3]Dekker, J. C. E. and Myhill, J., Retraceable sets, Canadian Journal of Mathematics, vol. 10 (1958), pp. 357–373.CrossRefGoogle Scholar
[4]Hummel, T., Effective versions of Ramsey's theorem, Ph.D. thesis, University of Illinois at Urbana-Champaign, Urbana, Illinois, 1993.Google Scholar
[5]Hummel, T. and Jockusch, C. G. Jr., Generalized cohesiveness, this Journal, vol. 64 (1999), pp. 489–516.Google Scholar
[6]Jockusch, C. G. Jr., Ramsey's Theorem and recursion theory, this Journal, vol. 37 (1972), pp. 268–280.Google Scholar
[7]Jockusch, C. G. Jr., and Soare, R., classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173 (1972), pp. 33–56.Google Scholar
[8]Jockusch, C. G. Jr., and Stephan, F., A cohesive set which is not high, Mathematical Logic Quarterly, vol. 39 (1993), pp. 515–530.CrossRefGoogle Scholar
[9]Jockusch, C. G. Jr., and Stephan, F., Correction to “A cohesive set which is not high”, Mathematical Logic Quarterly, vol. 43 (1997), p. 569.CrossRefGoogle Scholar
[10]Lawton, L., Ph.D. thesis in progress at the University of Illinois at Urbana-Champaign.Google Scholar
[11]Miller, W. and Martin, D. A., The degree of hyperimmune sets, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 14 (1968), pp. 159–166.CrossRefGoogle Scholar
[12]Ramsey, F. P., On a problem in formal logic, Proceedings of the London Mathematical Society, vol. 30 (1930), pp. 264–286.CrossRefGoogle Scholar
[13]Rogers, H. Jr., Theory of recursive functions and effective computability, The MIT Press, Cambridge, 1987.Google Scholar
[14]Seetapun, D. and Slaman, T., On the strength of Ramsey's Theorem, Notre Dame Journal of Formal Logic, vol. 36 (1995), pp. 570–582.CrossRefGoogle Scholar
[15]Soare, R. I., Recursively enumerable sets and degrees, Springer-Verlag, Berlin, 1987.CrossRefGoogle Scholar
[16]Specker, E., Ramsey's Theorem does not hold in recursive set theory, Logic colloquium '69, North Holland, Amsterdam, 1971, pp. 439–442.CrossRefGoogle Scholar