Index Sets in Recursive Combinatorics

  • William Gasarch
  • Georgia Martin
Part of the Progress in Computer Science and Applied Logic book series (PCS, volume 12)


Many theorems in infinite combinatorics have noneffective proofs. Nerode’s recursive mathematics program [10] involves looking at noneffective proofs and seeing if they can be made effective. The framework is recursion-theoretic. Typically, if a theorem has a noneffective proof, one would find a ‘recursive version’ of it and see if it is true. Usually the recursive version is false, hence the original proof is necessarily noneffective.


Partial Order Bipartite Graph Greedy Algorithm Turing Machine Line Graph 
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  1. [1]
    Beigel, R. and W.I. Gasarch [1989], On the complexity of finding the chromatic number of a recursive graph I: The bounded case. Annals of Pure and Applied Logic, 45(1), 1–38.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Even, S., A. Selman and Y. Yacobi [1984], The complexity of promise problems with applications to public-key cryptography. Information and Control, 61(2), 159–173.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Hall, M. [1948], Distinct representatives of subsets. Bull. of the American Math. Soc. 54, 922–926.CrossRefzbMATHGoogle Scholar
  4. [4]
    Harel, D. [1991], Hamiltonian paths in infinite graphs. Israel J. Mathematics 76, 317–336. (Shorter version appeared in STOC 1991.)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Jockusch, Jr. C.G. [1972], Ramsey’s theorem and recursion theory. J. Symbolic Logic 37, 268–280.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Kierstead, H.A. [1981], An effective version of Dilworth’s theorem. Trans. of the AMS 268, 63–77.MathSciNetzbMATHGoogle Scholar
  7. [7]
    Kierstead, H.A. [1986], Recursive ordered sets. In Combinatorics and Ordered Sets, vol. 57 of Contemporary Mathematics. American Mathematical Society.Google Scholar
  8. [8]
    König, D. [1926], Sur les correspondances multivoques des ensembles, Fundamenta Mathematicae 26, 114–130.Google Scholar
  9. [9]
    Manaster, A. and J. Rosenstein [1972], Effective matchmaking. Proc. of the London Math. Soc. 25, 615–654.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Metakides, G. and A. Nerode [1979], Effective content of field theory. Annals of Mathematical Logic 17, 289–320.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Soare, R.I. [1987], Recursively Enumerable Sets and Degrees. Perspectives in Mathematical Logic. Springer-Verlag, Berlin.Google Scholar
  12. [12]
    Specker, E. [1971], Ramsey’s Theorem does not hold in recursive set theory. In Logic Colloquium; 1969 Manchester, 439–442.Google Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • William Gasarch
    • 1
  • Georgia Martin
    • 2
  1. 1.Dept. of Computer Science and Institute for Advanced StudiesUniversity of MarylandCollege ParkUSA
  2. 2.Dept. of MathematicsUniversity of MarylandCollege ParkUSA

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