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DOMINATION AND REGULARITY

Part of: Model theory

Published online by Cambridge University Press:  19 January 2021

ANAND PILLAY
ANAND PILLAY*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF NOTRE DAMENOTRE DAME, IN, USA E-mail: [email protected]
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Abstract

We discuss the close relationship between structural theorems in (generalized) stability theory, and graph regularity theorems.

Keywords

Graph regularitystabilityNIP

MSC classification

Primary: 03C45: Classification theory, stability and related concepts
Secondary: 03C20: Ultraproducts and related constructions
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 26 , Issue 2 , June 2020 , pp. 103 - 117
Copyright
© The Association for Symbolic Logic 2021

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References

Alon, N., Fischer, E., and Newman, I., Efficient testing of bipartite graphs for forbidden induced subgraphs . SIAM Journal on Computing , vol. 37 (2007), pp. 959–976.CrossRefGoogle Scholar
Chernikov, A. and Starchenko, S., Regularity lemma for distal structures . Journal of the European Mathematical Society , vol. 20 (2018), pp. 2437–2466.CrossRefGoogle Scholar
Chernikov, A. and Starchenko, S., Definable regularity lemmas for NIP hypergraphs, preprint, 2016.Google Scholar
Conant, G. and Pillay, A., Pseudofinite groups and VC-dimension, preprint, 2018.Google Scholar
Conant, G., Pillay, A., and Terry, C., A group version of stable regularity . Proceedings of Cambridge Philosophical Society , vol. 24 (2018), pp. 1–9.Google Scholar
Conant, G., Pillay, A., and Terry, C., Structure and regularity for subsets of groups with finite VC-dimension . Journal of the European Mathematical Society , to appear.Google Scholar
Fox, J., Gromov, M., Lafforgue, V., Naor, A., and Pach, J., Overlap properties of geometric expanders . Journal für die reine und angewandte Mathematik , vol. 671 (2012), pp. 49–83.Google Scholar
Fox, J., Pach, J., and Suk, A., Erdos-Hajnal conjecture for graphs with bounded VC-dimension . Discrete and Computational Geometry , vol. 61 (2019), pp. 809–829.CrossRefGoogle Scholar
Hrushovski, E., Peterzil, Y., and Pillay, A., Groups, measures, and the NIP . Journal of the American Mathematical Society , vol. 21 (2008), pp. 563–595.CrossRefGoogle Scholar
Hrushovski, E. and Pillay, A., ON NIP and invariant measures . Journal of the European Mathematical Society , vol. 13 (2011), pp. 1005–10061.CrossRefGoogle Scholar
Hrushovski, E., Pillay, A., and Simon, P., Generically stable and smooth measures in NIP theories . Transactions of the American Mathematical Society , vol. 365 (2013), pp. 2341–2366.CrossRefGoogle Scholar
Keisler, H. J., Measures and forking . Annals of Pure and Applied Logic , vol. 45 (1987), pp. 119–169.CrossRefGoogle Scholar
Lovasz, L. and Szegedy, B., Regularity partitions and the topology of graphons , An Irregular Mind: Szemerédi is 70 (Bárány, S., editor), Springer, New York, 2010, pp. 415–446.CrossRefGoogle Scholar
Malliaris, M. and Pillay, A., The stable regularity lemma, revisited . Proceedings of the American Mathematical Society , vol. 244 (2016), pp. 1761–1765.Google Scholar
Malliaris, M. and Shelah, S., Regularity lemmas for stable graphs . Transactions of the American Mathematical Society , vol. 366 (2014), pp. 1551–1585.CrossRefGoogle Scholar
Pillay, A., Geometric Stability Theory , Oxford University Press, Oxford, 1996.Google Scholar
Simon, P., A Guide to NIP Theories , Lecture Notes in Logic, vol. 44, ASL-Cambridge University Press, Cambridge, 2015.CrossRefGoogle Scholar
Simon, P., A note on “Regularity theorem for distal structures” . Proceedings of the American Mathematical Society , vol. 144 (2016), pp. 3573–3578.CrossRefGoogle Scholar
Terry, C. and Wolf, J., Stable arithmetic regularity in the finite field model . Bulletin of the London Mathematical Society , vol. 51 (2019), pp. 70–88.CrossRefGoogle Scholar