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EXISTENCE OF MODELING LIMITS FOR SEQUENCES OF SPARSE STRUCTURES

Published online by Cambridge University Press:  07 March 2019

JAROSLAV NEŠETŘIL  and
PATRICE OSSONA DE MENDEZ
JAROSLAV NEŠETŘIL
Affiliation:
COMPUTER SCIENCE INSTITUTE OF CHARLES UNIVERSITY (IUUK AND ITI) MALOSTRANSKÉ NÁM.25, 11800PRAHA 1, CZECH REPUBLIC E-mail: [email protected]
PATRICE OSSONA DE MENDEZ
Affiliation:
CENTRE D’ANALYSE ET DE MATHÉMATIQUES SOCIALES (CNRS, UMR 8557) 190-198 AVENUE DE FRANCE, 75013 PARIS, FRANCE and COMPUTER SCIENCE INSTITUTE OF CHARLES UNIVERSITY (IUUK) MALOSTRANSKÉ NÁM.25, 11800PRAHA 1, CZECH REPUBLIC E-mail: [email protected]
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Abstract

A sequence of graphs is FO-convergent if the probability of satisfaction of every first-order formula converges. A graph modeling is a graph, whose domain is a standard probability space, with the property that every definable set is Borel. It was known that FO-convergent sequence of graphs do not always admit a modeling limit, but it was conjectured that FO-convergent sequences of sufficiently sparse graphs have a modeling limits. Precisely, two conjectures were proposed:

  1. 1. If a FO-convergent sequence of graphs is residual, that is if for every integer d the maximum relative size of a ball of radius d in the graphs of the sequence tends to zero, then the sequence has a modeling limit.

  2. 2. A monotone class of graphs ${\cal C}$ has the property that every FO-convergent sequence of graphs from ${\cal C}$ has a modeling limit if and only if ${\cal C}$ is nowhere dense, that is if and only if for each integer p there is $N\left( p \right)$ such that no graph in ${\cal C}$ contains the pth subdivision of a complete graph on $N\left( p \right)$ vertices as a subgraph.

In this article we prove both conjectures. This solves some of the main problems in the area and among others provides an analytic characterization of the nowhere dense–somewhere dense dichotomy.

Keywords

Primary 03C3003C9805C99Secondary 03C4528E1503C50first-order propertieslimit statisticsconvergence of graphsstructural limittotally Borel graphmodeling
Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 2 , June 2019 , pp. 452 - 472
Copyright
Copyright © The Association for Symbolic Logic 2019 

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References

REFERENCES

Adler, H. and Adler, I., Interpreting nowhere dense graph classes as a classical notion of model theory. European Journal of Combinatorics, vol. 36 (2014), pp. 322–330.Google Scholar
Aldous, D., Representations for partially exchangeable arrays of random variables. Journal of Multivariate Analysis, vol. 11 (1981), pp. 581–598.Google Scholar
Benjamini, I. and Schramm, O., Recurrence of distributional limits of finite planar graphs. Electronic Journal of Probability, vol. 6 (2001), no. 23, p. 13.Google Scholar
Borgs, C., Chayes, J. T., and Lovász, L, Moments of two-variable functions and the uniqueness of graph limits. Geometric and Functional Analysis, vol. 19 (2012), no. 6, pp. 1597–1619.Google Scholar
Borgs, C., Chayes, J. T., Lovász, L., Sós, V. T., Szegedy, B., and Vesztergombi, K., Graph limits and parameter testing, Proceedings of the 38th Annual ACM Symposium on Theory of Computing, Association for Computing Machinery, New York, 2006, pp. 261–270.Google Scholar
Borgs, C., Chayes, J. T., Lovász, L., Sós, V. T., and Vesztergombi, K., Counting graph homomorphisms, Topics in Discrete Mathematics (Klazar, M., Kratochvíl, J., Loebl, M., Matoušek, J., Thomas, R., and Valtr, P., editors), Algorithms and Combinatorics, vol. 26, Springer Verlag, Berlin, 2006, pp. 315–371.Google Scholar
Borgs, C., Chayes, J. T., Lovász, L., Sós, V. T., and Vesztergombi, K., Convergent sequences of dense graphs I: Subgraph frequencies, metric properties and testing. Advances in Mathematics, vol. 219 (2008), no. 6, pp. 1801–1851.Google Scholar
Conley, C. T., Kechris, A. S., and Tucker-Drob, R. D., Ultraproducts of measure preserving actions and graph combinatorics. Ergodic Theory and Dynamical Systems, vol. 33 (2012), no. 2, pp. 334–374.Google Scholar
Elek, G., Note on limits of finite graphs. Combinatorica, vol. 27 (2007), pp. 503–507.Google Scholar
Elek, G. and Szegedy, B., Limits of hypergraphs, removal and regularity lemmas. A nonstandard approach, 2007, arXiv:0705.2179v1 [math.CO].Google Scholar
Friedman, H. M., Addendum to on the logic of measure and category I, Ohio State University, manuscript, 1979.Google Scholar
Friedman, H. M., Borel structures and mathematics, manuscript, 1979.Google Scholar
Gaifman, H., On local and nonlocal properties, Proceedings of the Herbrand Symposium, Logic Colloquium ’81 (Stern, J., editor), Studies in Logic and the Foundations of Mathematics, vol. 107, Elsevier, Amsterdam, 1982, pp. 105–135.Google Scholar
Gajarský, J., Hliněný, P., Kaiser, T., Kráľ, D., Kupec, M., Obdržálek, J., Ordyniak, S., and Tůma, V., First-order limits of sparse graphs: Plane trees and path-width, http://arxiv.org/abs/1504.08122v1, 2015.Google Scholar
Grohe, M., Kreutzer, S., and Siebertz, S., Deciding first-order properties of nowhere dense graphs, Proceedings of the 46th Annual ACM Symposium on Theory of Computing, Association for Computing Machinery, New York, 2014, pp. 89–98.Google Scholar
Hoover, D., Relations on probability spaces and arrays of random variables, Technical report, Institute for Advanced Study, Princeton, NJ, 1979.Google Scholar
Kardoš, F., Kráľ, D., Liebenau, A., and Mach, L., First-order convergence of matroids, 2015, arXiv:1501.06518v1 [math.CO].Google Scholar
Laskowski, M. C., Vapnik-Chervonenkis classes of definable sets. The Journal of the London Mathematical Society, vol. 45 (1992), no. 2, pp. 377–384.Google Scholar
Leitgeb, H., Probability in logic, The Oxford Handbook of Probability and Philosophy (Hájek, A. and Hitchcock, C., editors), Oxford University Press, Oxford, 2016, pp. 681–704.Google Scholar
Łoś, J., Quelques remarques, théorèmes et problèmes sur les classes définissables d’algèbres, Mathematical Interpretation of Formal Systems (Brouwer, L. E. J., Beth, E. W., and Heyting, A., editors), Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1955, pp. 98–113.Google Scholar
Lovász, L., Large Networks and Graph Limits, Colloquium Publications, vol. 60, American Mathematical Society, Providence, RI, 2012.Google Scholar
Lovász, L. and Szegedy, B., Limits of dense graph sequences. Journal of Combinatorial Theory, Series B, vol. 96 (2006), pp. 933–957.Google Scholar
Lovász, L. and Szegedy, B., Regularity partitions and the topology of graphons, An Irregular Mind (Szemerédi is 70) (Bárány, I. and Solymosi, J., editors), Bolyai Society Mathematical Studies, vol. 21, Springer, Berlin, 2010, pp. 415–446.Google Scholar
Nešetřil, J. and Ossona de Mendez, P., Grad and classes with bounded expansion I. Decompositions. European Journal of Combinatorics, vol. 29 (2008), no. 3, pp. 760–776.Google Scholar
Nešetřil, J. and Ossona de Mendez, P., Grad and classes with bounded expansion II. Algorithmic aspects. European Journal of Combinatorics, vol. 29 (2008), no. 3, pp. 777–791.Google Scholar
Nešetřil, J. and Ossona de Mendez, P., Grad and classes with bounded expansion III. Restricted graph homomorphism dualities. European Journal of Combinatorics, vol. 29 (2008), no. 4, pp. 1012–1024.Google Scholar
Nešetřil, J. and Ossona de Mendez, P., First-order properties on nowhere dense structures, this Journal, vol. 75 (2010), no. 3, pp. 868–887.Google Scholar
Nešetřil, J. and Ossona de Mendez, P., On nowhere dense graphs. European Journal of Combinatorics, vol. 32 (2011), no. 4, pp. 600–617.Google Scholar
Nešetřil, J. and Ossona de Mendez, P., A model theory approach to structural limits. Commentationes Mathematicæ Universitatis Carolinæ, vol. 53 (2012), no. 4, pp. 581–603.Google Scholar
Nešetřil, J. and Ossona de Mendez, P., Sparsity (graphs, structures, and algorithms), Algorithms and Combinatorics, vol. 28, Springer, Berlin, 2012, p. 465.Google Scholar
Nešetřil, J. and Ossona de Mendez, P., First-order limits, an analytical perspective. European Journal of Combinatorics, vol. 52 Part B (2016), pp. 368–388. Recent Advances in Graphs and Analysis (special issue).Google Scholar
Nešetřil, J. and Ossona de Mendez, P., Modeling limits in hereditary classes: Reduction and application to trees. Electronic Journal of Combinatorics, vol. 23 (2016), no. 2, p. #P2.52.Google Scholar
Nešetřil, J. and Ossona de Mendez, P., Structural sparsity. Uspekhi Matematicheskikh Nauk, vol. 71 (2016), no. 1, pp. 85–116. (Russian Mathematical Surveys, vol. 71, no. 1, pp. 79–107).Google Scholar
Nešetřil, J. and Ossona de Mendez, P., Limits of mappings. European Journal of Combinatorics, vol. 66 (2017), pp. 145–159. Selected papers of EuroComb15 (special issue).Google Scholar
Nešetřil, J. and Ossona de Mendez, P., A Unified Approach to Structural Limits (With Application to the Study of Limits of Graphs with Bounded Tree-Depth), Memoirs of the American Mathematical Society, to appear.Google Scholar
Nešetřil, J. and Ossona de Mendez, P., Approximation of mappings. Building Bridges II, 2018, submitted.Google Scholar
Steinhorn, C., Borel structures for first-order and extended logics, Harvey Friedman’s Research on the Foundations of Mathematics (Sčědrov, A., Harrington, L. A., Morley, M. D., and Simpson, S. G., editors), Studies in Logic and the Foundations of Mathematics, vol. 117, Elsevier, Amsterdam, 1985, pp. 161–178.Google Scholar
Steinhorn, C., Chapter XVI Borel structures and measure and category logics, Model-Theoretic Logics (Barwise, J. and Feferman, S., editors), Perspectives in Mathematical Logic, vol. 8, Springer-Verlag, New York, 1985, pp. 579–596.Google Scholar