Combinatorial bounds on relational complexity

  • David Hartman
  • Jan Hubička
  • Jaroslav Nešetřil
Conference paper
Part of the CRM Series book series (PSNS, volume 16)

Abstract

An ultrahomogeneous structure is a (finite or countable) relational structure for which every partial isomorphism between finite substructures can be extended to a global isomorphism. This very strong symmetry condition implies that there are just a few ultrahomogeneous structures. For example, by [14], there are just countably many ultrahomogeneous undirected graphs. The classification program is one of the celebrated lines of research in the model theory, see [4, 15]. Various measures were introduced in order to modify a structure to an ultrahomogeneous one. A particularly interesting measure is the minimal arity of added relations (i.e. the minimal arity of an extension or lift) which suffice to produce an ultrahomogeneous structure. If these added relations are not changing the automorphism group then the problem is called the relational complexity and this is the subject of this paper. In the context of permutation groups, the relational complexity was defined in [5] and was recently popularized by Cherlin [2,3]. We determine the relational complexity of one of the most natural class of structures (the class of structures defined by forbidden homomorphisms). This class has a (countably) universal structure [6]. As a consequence of our main result (Theorem 3.1) we strengthen this by determining its relational complexity. Although formulated in the context of model theory this result has a combinatorial character. Full details will appear in [9].

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    P. J. Cameron, The age of a relational structure, In: “Directions in Infinite Graph Theory and Combatorics”, R. Diestel (ed.), Topic in Discrete Math. 3, North-Holland, Amsterdam (1992), 49–67.Google Scholar
  2. [2]
    G. L. Cherlin, Finite Groups and Model Theory, In: “Proceedings of 2nd workshop on homogeneous structures”, D. Hartman (ed.), Matfyz press (2012), 6–8.Google Scholar
  3. [3]
    G. L. Cherlin, On the relational complexity of a finite permutation group, in preparation, available at http://www.math.rutgers.edu/∼cherlin/Paper/inprep.html/∼cherlin/Paper/inprep.html.
  4. [4]
    G. L. Cherlin, “The classification of countable homogeneous directed graphs and countable homogeneous n-tournaments”, Memoirs Amer. Math. Soc. 621, American Mathematical Society, Providence, RI (1998).Google Scholar
  5. [5]
    G. L. Cherlin, G. Martin and D. Saracino, Arities of permutation groups: Wreath products and k-sets, J. Combinatorial Theory, Ser. A 74 (1996), 249–286.CrossRefMATHMathSciNetGoogle Scholar
  6. [6]
    G. L. Cherlin, S. Shelah and N. Shi, Universal Graphs with Forbidden Subgraphs and Algebraic Closure, Advances in Applied Mathematics 22 (1999), 454–491.CrossRefMATHMathSciNetGoogle Scholar
  7. [7]
    J. Covington, Homogenizable Relational Structures, Illinois J. Mathematics 34(4) (1990), 731–743.MATHMathSciNetGoogle Scholar
  8. [8]
    P. L. Erdős, D. Pálvölgyi, C. Tardif and G. Tardos, On infinite-finite tree-duality pairs of relational structures, arXiv:1207.4402v1 (submitted) (2012).Google Scholar
  9. [9]
    D. Hartman, J. Hubička, J. Nešetřil, Complexities of relational structures, Math. Slovaca, accepted for publication.Google Scholar
  10. [10]
    W. Hodges, “Model Theory”, Cambridge University Press, 1993.Google Scholar
  11. [11]
    J. Hubička and J. Nešetřil, Homomorphism and embedding universal structures for restricted classes, arXiv:0909.4939.Google Scholar
  12. [12]
    J. Hubička and J. Nešetřil, Universal structures with forbidden homomorphisms, arXiv:0907.4079, to appear in J. Väänänen Festschrift, Ontos.Google Scholar
  13. [13]
    A. S. Kechris, V. G. Pestov and S. Todorčevič, Fraïssé Limits, Ramsey Theory, and Topological Dynamics of Automorphism Groups, Geom. Funct. Anal. 15 (2005), 106–189.CrossRefMATHMathSciNetGoogle Scholar
  14. [14]
    A. H. Lachlan and R. E. Woodroow, Countable ultrahomogeneous graphs, Trans. Amer. Math. Soc. 284(2) (1984), 431–461.CrossRefMATHMathSciNetGoogle Scholar
  15. [15]
    A. H. Lachlan, Homogeneous Structures, In: “Proc. of the ICM 1986”, AMS, Providence, 1987, 314–321.Google Scholar
  16. [16]
    J. Nešetřil, For graphs there are only four types of hereditary Ramsey Classes, J. Combin. Theory B 46(2) (1989), 127–132.CrossRefMATHGoogle Scholar
  17. [17]
    J. Nešetřil, Ramsey Classes and Homogeneous Structures, Combinatorics, Probablity and Computing (2005) 14, 171–189.CrossRefMATHGoogle Scholar

Copyright information

© Scuola Normale Superiore Pisa 2013

Authors and Affiliations

  • David Hartman
    • 1
  • Jan Hubička
    • 1
  • Jaroslav Nešetřil
    • 1
  1. 1.Czech Republic Computer Science Institute of Charles UniversityCzech Republic

Personalised recommendations