Hostname: page-component-7f64f4797f-l842n Total loading time: 0 Render date: 2025-11-05T20:49:27.764Z Has data issue: false hasContentIssue false
  • English
  • Français

NOTES ON THE STABLE REGULARITY LEMMA

Published online by Cambridge University Press:  06 December 2021

M. MALLIARIS  and
S. SHELAH Open the ORCID record for S. SHELAH [Opens in a new window]
M. MALLIARIS
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CHICAGO 5734 S. UNIVERSITY AVENUE CHICAGO, IL 60637, USA E-mail: mem@math.uchicago.edu
S. SHELAH
Affiliation:
EINSTEIN INSTITUTE OF MATHEMATICS EDMOND J. SAFRA CAMPUS, GIVAT RAM THE HEBREW UNIVERSITY OF JERUSALEM JERUSALEM, 91904, ISRAEL and DEPARTMENT OF MATHEMATICS HILL CENTER – BUSCH CAMPUS, RUTGERS THE STATE UNIVERSITY OF NEW JERSEY 110 FRELINGHUYSEN ROAD, PISCATAWAY, NJ 08854, USA E-mail: shelah@math.huji.ac.il URL: http://shelah.logic.at
Get access Rights & Permissions [Opens in a new window]

Abstract

This is a short expository account of the regularity lemma for stable graphs proved by the authors, with some comments on the model theoretic context, written for a general logical audience.

Information

Type
Articles
Information
Bulletin of Symbolic Logic , Volume 27 , Issue 4 , December 2021 , pp. 415 - 425
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Association for Symbolic Logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Ackerman, N., Freer, C., and Patel, R., Stable regularity for relational structures, preprint, 2014, arXiv:1712.09305.Google Scholar
Alon, N., Duke, R., Leffman, H., Rödl, V., and Yuster, R., The algorithmic aspects of the regularity lemma. Foundations of Computer Science, vol. 33 (1992), pp. 479–481.Google Scholar
Alon, N., Duke, R., Leffman, H., Rödl, V., and Yuster, R., The algorithmic aspects of the regularity lemma. Journal of Algorithms, vol. 16 (1994), pp. 80–109.CrossRefGoogle Scholar
Alon, N., Fox, J., and Zhao, Y., Efficient arithmetic regularity and removal lemmas for induced bipartite patterns. Discrete Analysis, vol. 3 (2019), 14 pp.Google Scholar
Conlon, D. and Fox, J., Bounds for graph regularity and removal lemmas. Geometric and Functional Analysis, vol. 22 (2012), pp. 1192–1256.CrossRefGoogle Scholar
Gowers, W. T., Lower bounds of tower type for Szemerédi’s uniformity lemma. Geometric and Functional Analysis, vol. 7 (1997), pp. 322–337.CrossRefGoogle Scholar
Hodges, W., Model Theory, Encyclopedia of Mathematics, Cambridge University Press, Cambridge, 1993.CrossRefGoogle Scholar
Komlós, J. and Simonovits, M., Szemerédi’s Regularity Lemma and its applications in graph theory , Combinatorics: Paul Erdös is Eighty, (Miklós, D., Sós, V. T. and Szönyi, T., editors), Bolyai Society Mathematical Studies, vol. 2, Keszthely, Hungary, 1996, pp. 295–352.Google Scholar
Lovász, L. and Szegedy, B., Regularity partitions and the topology of graphons , An Irregular Mind (Szemerédi is 70), Bolyai Society Mathematical Studies, vol. 21, Keszthely, Hungary, 2010, pp. 415–446.CrossRefGoogle Scholar
Malliaris, M., Persistence and regularity in unstable model theory , Ph.D. thesis, Berkeley, 2009.Google Scholar
Malliaris, M., Edge distribution and density in the characteristic sequence. Annals of Pure and Applied Logic, vol. 162 (2010), no. 1, pp. 1–19.CrossRefGoogle Scholar
Malliaris, M. and Shelah, S., Regularity lemmas for stable graphs. Transactions of the American Mathematical Society, vol. 366 (2014), pp. 1551–1585. First public version: arXiv:1102.3904 (Feb 2011).CrossRefGoogle Scholar
Malliaris, M. and Shelah, S., Keisler’s order is not simple (and simple theories may not be either). Advances in Mathematics 2021, Paper No. 108036, 94 pp.CrossRefGoogle Scholar
Malliaris, M. and Terry, C., On unavoidable induced subgraphs in large prime graphs. Journal of Graph Theory, vol. 88 (2018), no. 2, pp. 255–270.CrossRefGoogle Scholar
Shelah, S., Classification Theory and the Number of Non-isomorphic Models, first ed., North-Holland, Amsterdam, 1978, rev. ed. 1990.Google Scholar
Szemerédi, E., On sets of integers containing no $k$ elements in arithmetic progression. Acta Arithmetica, vol. 27 (1975), pp. 199–245.CrossRefGoogle Scholar
Szemerédi, E., Regular partitions of graphs , Colloques Internationaux C.N.R.S. No. 260, Problèmes Combinatoires et Théorie Des Graphes, Univ. Orsay, Orsay, 1976, pp. 399–401.Google Scholar
Terry, C. and Wolf, J., Stable arithmetic regularity in the finite-field model. The Bulletin of the London Mathematical Society, vol. 51 (2019), no. 1, pp. 70–88.CrossRefGoogle Scholar
Vapnik, V. and Chervonenkis, A., On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and its Applications , vol. XVI (1971), no. 2, pp. 264–280.CrossRefGoogle Scholar