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Minimal functions on the random graph

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Abstract

We show that there is a system of 14 non-trivial finitary functions on the random graph with the following properties: Any non-trivial function on the random graph generates one of the functions of this system by means of composition with automorphisms and by topological closure, and the system is minimal in the sense that no subset of the system has the same property. The theorem is obtained by proving a Ramsey-type theorem for colorings of tuples in finite powers of the random graph, and by applying this to find regular patterns in the behavior of any function on the random graph. As model-theoretic corollaries of our methods we rederive a theorem of Simon Thomas classifying the first-order closed reducts of the random graph, and prove some refinements of this theorem; also, we obtain a classification of the maximal reducts closed under primitive positive definitions, and prove that all reducts of the random graph are model-complete.

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References

  1. F. G. Abramson and L. Harrington, Models without indiscernibles, Journal of Symbolic Logic 43 (1978), 572–600.

    Article  MATH  MathSciNet  Google Scholar 

  2. M. Bodirsky and H. Chen, Oligomorphic clones, Algebra Universalis 57 (2007), 109–125.

    Article  MATH  MathSciNet  Google Scholar 

  3. M. Bodirsky, H. Chen, J. Kára and T. von Oertzen, Maximal infinite-valued constraint languages, Theoretical Computer Science 410 (2009), 1684–1693. A preliminary version appeared at ICALP’07.

    Article  MATH  MathSciNet  Google Scholar 

  4. M. Bodirsky, H. Chen and M. Pinsker, The reducts of equality up to primitive positive interdefinability, Journal of Symbolic Logic 75 (2010), 1249–1292.

    Article  MATH  MathSciNet  Google Scholar 

  5. M. Bodirsky, P. Jonsson and T. von Oertzen, Horn versus full first-order: a complexity dichotomy for algebraic constraint satisfaction problems, Journal of Logic and Computation 22 (2010), 643–660.

    Article  Google Scholar 

  6. M. Bodirsky and J. Kára, The complexity of equality constraint languages, Theory of Computing Systems 3 (2008), 136–158. A conference version appeared in the proceedings of CSR’06.

    Article  Google Scholar 

  7. M. Bodirsky and J. Kára, The complexity of temporal constraint satisfaction problems, Journal of the ACM 57 (2009), 41 pp. An extended abstract appeared in the proceedings of STOC’08.

  8. M. Bodirsky and J. Nešetril, Constraint satisfaction with countable homogeneous templates, Journal of Logic and Computation 16 (2006), 359–373.

    Article  MATH  MathSciNet  Google Scholar 

  9. M. Bodirsky and M. Pinsker, Topological Birkhoff, Transactions of the American Mathematical Society, to appear. Preprint arxiv.org/abs/1203.1876.

  10. M. Bodirsky and M. Pinsker, Reducts of Ramsey structures, in Model Theoretic Methods in Finite Combinatorics, Contemporary Mathematics, Vol. 558, American Mathematical Society, Providence, RI, 2011, pp. 489–519.

    Chapter  Google Scholar 

  11. M. Bodirsky and M. Pinsker, Schaefer’s theorem for graphs, in STOC’ 11—Proceedings of the 43rd ACM Symposium on Theory of Computing, ACM, New York, 2011, pp. 655–664. Preprint of the long version available at arxiv.org/abs/1011.2894.

    Google Scholar 

  12. M. Bodirsky, M. Pinsker and T. Tsankov, Decidability of definability, in 26th Annual IEEE Symposium on Logic in Computer Science—LICS 2011, IEEE Computer Society, Los Alamitos, CA, 2011, pp. 321–328. Preprint of full journal version available from arxiv.org/abs/1012.2381.

    Chapter  Google Scholar 

  13. A. Bonato and D. Delić, The monoid of the random graph, Semigroup Forum 61 (2000), 138–148.

    Article  MATH  MathSciNet  Google Scholar 

  14. A. Bonato, D. Delić and I. Dolinka, All countable monoids embed into the monoid of the infinite random graph, Discrete Mathematics 310 (2010), 373–375.

    Article  MATH  MathSciNet  Google Scholar 

  15. A. A. Bulatov, A dichotomy theorem for constraint satisfaction problems on a 3-element set, Journal of the ACM 53 (2006), 66–120.

    Article  MathSciNet  Google Scholar 

  16. P. J. Cameron, Transitivity of permutation groups on unordered sets, Mathematische Zeitschrift 148 (1976), 127–139.

    Article  MATH  MathSciNet  Google Scholar 

  17. P. J. Cameron, Oligomorphic Permutation Groups, Cambridge University Press, Cambridge, 1990.

    Book  MATH  Google Scholar 

  18. P. J. Cameron, The random graph, Algorithms and Combinatorics 14 (1997), 333–351.

    Article  Google Scholar 

  19. P. J. Cameron, The random graph revisited, in Proceedings of the European Congress of Mathematics, Vol. 201, Birkhäuser, Basel, 2001, pp. 267–274.

    Google Scholar 

  20. D. Delić and I. Dolinka, The endomorphism monoid of the random graph has uncountably many ideals, Semigroup Forum 69 (2004), 75–79.

    Article  MATH  MathSciNet  Google Scholar 

  21. M. Goldstern and M. Pinsker, A survey of clones on infinite sets, Algebra Universalis 59 (2008), 365–403.

    Article  MATH  MathSciNet  Google Scholar 

  22. W. Hodges, A Shorter Model Theory, Cambridge University Press, Cambridge, 1997.

    MATH  Google Scholar 

  23. M. Junker and M. Ziegler, The 116 reducts of (ℚ,<,a), Journal of Symbolic Logic 74 (2008), 861–884.

    Article  MathSciNet  Google Scholar 

  24. J. Nešetřil and V. Rödl, Partitions of finite relational and set systems, Journal of Combinatorial Theory. Series A 22 (1977), 289–312.

    MATH  MathSciNet  Google Scholar 

  25. J. Nešetřil, Ramsey theory, in Handbook of Combinatorics, Vol. 1, 2, Elsevier, Amsterdam, 1995, pp. 1331–1403.

    Google Scholar 

  26. J. Nešetřil and V. Rödl, Ramsey classes of set systems, Journal of Combinatorial Theory. Series A 34 (1983), 183–201.

    Article  MathSciNet  Google Scholar 

  27. J. Nešetřil and V. Rödl, The partite construction and Ramsey set systems, Discrete Mathematics 75 (1989), 327–334.

    Article  MATH  MathSciNet  Google Scholar 

  28. M. Pinsker, More sublattices of the lattice of local clones, Order 27 (2010), 353–364.

    Article  MATH  MathSciNet  Google Scholar 

  29. R. Pöschel and L. A. Kalužnin, Funktionen- und Relationenalgebren, Ein Kapital der diskreten Mathematik, Mathenatische Monographien, Vol. 15, Deutscher Verlag der Wissenschaften, Berlin, 1979.

    Book  Google Scholar 

  30. T. J. Schaefer, The complexity of satisfiability problems, in Conference Record of the Tenth Annual ACM Symposium on Theory of Computing (San Diego, CA, 1978), ACM, New York, 1978, pp. 216–226.

    Google Scholar 

  31. Á. Szendrei, Clones in Universal Algebra, Séminaire de Mathématiques Supérieures, Vol. 99, Presses de l’Université de Montréal, Montréal, QC, 1986.

    MATH  Google Scholar 

  32. S. Thomas, Reducts of the random graph, Journal of Symbolic Logic 56 (1991), 176–181.

    Article  MATH  MathSciNet  Google Scholar 

  33. S. Thomas, Reducts of random hypergraphs, Annals of Pure and Applied Logic 80 (1996), 165–193.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Laboratoire d’Informatique (LIX), CNRS UMR 7161, École Polytechnique, 91128, Palaiseau, France

    Manuel Bodirsky

  2. Équipe de Logique Mathématique, Université Diderot-Paris, 7 UFR de Mathématiques, 75205, Paris Cedex 13, France

    Michael Pinsker

Authors
  1. Manuel Bodirsky
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  2. Michael Pinsker
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Correspondence to Manuel Bodirsky.

Additional information

The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013 Grant Agreement no. 257039). The second author is grateful for support through Erwin Schrödinger Fellowship J2742-N18 of the Austrian Science Fund and through an APART-fellowship of the Austrian Academy of Sciences.

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Bodirsky, M., Pinsker, M. Minimal functions on the random graph. Isr. J. Math. 200, 251–296 (2014). https://doi.org/10.1007/s11856-014-1042-y

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  • DOI: https://doi.org/10.1007/s11856-014-1042-y

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