The Seventh European Conference on Combinatorics, Graph Theory and Applications pp 573-578 | Cite as

# Combinatorial bounds on relational complexity

## 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.

### References

- [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]G. L. Cherlin,
*Finite Groups and Model Theory*, In: “Proceedings of 2^{nd}workshop on homogeneous structures”, D. Hartman (ed.), Matfyz press (2012), 6–8.Google Scholar - [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]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]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]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]J. Covington,
*Homogenizable Relational Structures*, Illinois J. Mathematics**34**(4) (1990), 731–743.MATHMathSciNetGoogle Scholar - [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]D. Hartman, J. Hubička, J. Nešetřil,
*Complexities of relational structures*, Math. Slovaca, accepted for publication.Google Scholar - [10]W. Hodges, “Model Theory”, Cambridge University Press, 1993.Google Scholar
- [11]J. Hubička and J. Nešetřil,
*Homomorphism and embedding universal structures for restricted classes*, arXiv:0909.4939.Google Scholar - [12]J. Hubička and J. Nešetřil,
*Universal structures with forbidden homomorphisms*, arXiv:0907.4079, to appear in J. Väänänen Festschrift, Ontos.Google Scholar - [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]A. H. Lachlan and R. E. Woodroow,
*Countable ultrahomogeneous graphs*, Trans. Amer. Math. Soc.**284**(2) (1984), 431–461.CrossRefMATHMathSciNetGoogle Scholar - [15]A. H. Lachlan,
*Homogeneous Structures*, In: “Proc. of the ICM 1986”, AMS, Providence, 1987, 314–321.Google Scholar - [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]J. Nešetřil,
*Ramsey Classes and Homogeneous Structures*, Combinatorics, Probablity and Computing (2005) 14, 171–189.CrossRefMATHGoogle Scholar