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Licensed Unlicensed Requires Authentication Published by De Gruyter May 22, 2015

Complexities of Relational Structures

David Hartman, Jan Hubička and Jaroslav Nešetřil
From the journal Mathematica Slovaca


The relational complexity, introduced by G. Cherlin, G. Martin, and D. Saracino, is a measure of ultrahomogeneity of a relational structure. It provides an information on minimal arity of additional invariant relations needed to turn given structure into an ultrahomogeneous one. The original motivation was group theory. This work focuses more on structures and provides an alternative approach. Our study is motivated by related concept of lift complexity studied by Hubička and Nešetřil.


[1] CAMERON, P. J.: The age of a relational structure. In: Directions in Infinite Graph Theory and Combinatorics (R. Diestel, ed.). Topic in Discrete Math. 3, North-Holland, Amsterdam, 1992, pp. 49-67.Search in Google Scholar

[2] CAMERON, P. J.-NEŠETŘIL, J.: Homomorphism-homogeneous relational structures, Combin. Probab. Comput. 15 (2006), 91-103.10.1017/S0963548305007091Search in Google Scholar

[3] CHERLIN, G.: The classification of countable homogeneous directed graphs and countable homogeneous n-tournaments, Mem. Amer. Math. Soc. 612 (1998), 1-161.Search in Google Scholar

[4] CHERLIN, G.: Sporadic homogeneous structures. In: The Gelfand Mathematical Seminars (1994-1999), Birkhäuser Boston, 2000, 15-48.10.1007/978-1-4612-1340-6_2Search in Google Scholar

[5] CHERLIN, G.: Two problems on homogeneous structures, revisited. In: Model Theoretic Methods in Finite Combinatorics (M. Grohe, J. A. Makowsky, eds.). Contemp. Math. 558, Amer. Math. Soc., Providence, RI, 2011, pp. 319-415.10.1090/conm/558/11055Search in Google Scholar

[6] CHERLIN, G.-MARTIN, G.-SARACINO, D.: Arities of permutation groups: Wreath products and k-sets, J. Combin. Theory Ser. A 74 (1996), 249-286.10.1006/jcta.1996.0050Search in Google Scholar

[7] CHERLIN, G. L.-SHELAH, S.-SHI, N.: Universal graphs with forbidden subgraphs and algebraic closure, Adv. in Appl. Math. 22 (1999), 454-491.10.1006/aama.1998.0641Search in Google Scholar

[8] COVINGTON, J.: Homogenizable relational structures, Illinois J. Math. 34 (1990), 731-743.10.1215/ijm/1255988065Search in Google Scholar

[9] FRÄISSÉ , R.: Sur certains relations qui généralisent l’ordre des nombres rationnels, C. R. Acad. Sci. Paris 237 (1953), 540-542.Search in Google Scholar

[10] GARDINER, A.: Homogeneous graphs, J. Combin. Theory Ser. B 20 (1976), 94-102.10.1016/0095-8956(76)90072-1Search in Google Scholar

[11] HODGES, W.: Model Theory, Cambridge University Press, Cambridge, 1993.Search in Google Scholar

[12] HUBIČKA, J.-NEŠETŘIL, J.: Universal structures with forbidden homomorphisms. In: Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics (A. Hirvonen, J. Kontinen, R. Kossak, A. Villaveces, eds.). Ontos Math. Logic Ser., De Gruyter, Berlin, 2015, pp. 241-264.Search in Google Scholar

[13] HUBIČKA, J.-NEŠETŘIL, J.: Homomorphism and embedding universal structures for restricted classes, J. Mult.-Valued Logic Soft Comput. (To appear).Search in Google Scholar

[14] JENKINSON, T.-SEIDEL, D.-TRUSS, J. K.: Countable homogeneous multipartite graphs, European J. Combin. 33 (2012), 82-109.10.1016/j.ejc.2011.04.004Search in Google Scholar

[15] KECHRIS, A. S.-PESTOV, V. G.-TODORČEVIČ, S.: Fräıssé limits, Ramsey theory, and topological dynamics of automorphism groups, Geom. Funct. Anal. 15 (2005), 106-189.10.1007/s00039-005-0503-1Search in Google Scholar

[16] KNIGHT, J.-LACHLAN, A. Shrinking, stretching, and codes for homogeneous structures. In: Classification Theory (J. Baldwin, ed.). Lecture Notes in Math. 1292, Sringer, New York, 1985.Search in Google Scholar

[17] LACHLAN, A. H.-WOODROW, A. H.: Countable ultra-homogeneous graphs, Trans. Amer. Math. Soc. 262 (1992), 51-94.10.1090/S0002-9947-1980-0583847-2Search in Google Scholar

[18] NEŠETŘIL, J.-TARDIF, C.: Duality theorems for finite structures (Characterising gaps and good characterisations), J. Combin. Theory Ser. B 80 (2000), 80-97.10.1006/jctb.2000.1970Search in Google Scholar

[19] MATOUŠEK, J-NEŠETŘIL, J.: Invitation to Discrete Mathematics, Oxford University Press, Oxford, 1998. 20] NEŠETŘIL, J.: For graphs there are only four types of hereditary Ramsey classes, J.Combin. Theory Ser. B 46 (1989), 127-132.10.1016/0095-8956(89)90038-5Search in Google Scholar

[21] NEŠETŘIL, J.: Ramsey classes and homogeneous structures, Combin. Probab. Comput. 14 (2005), 171-189.10.1017/S0963548304006716Search in Google Scholar

[22] ROSE, S. E.: Classification of Countable Homogeneous 2-Graphs. PhD Thesis, University of Leeds, 2011. Search in Google Scholar

Received: 2013-5-30
Accepted: 2013-6-6
Published Online: 2015-5-22
Published in Print: 2015-4-1

© Mathematical Institute Slovak Academy of Sciences

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