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A tight lower bound for Szemerédi’s regularity lemma

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Abstract

We determine the order of the tower height for the partition size in a version of Szemerédi’s regularity lemma. This addresses a question of Gowers.

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References

  1. N. Alon and F. R. K. Chung: Explicit construction of linear sized tolerant networks, Discrete Math. 72 (1988), 15–19.

    Article  MathSciNet  MATH  Google Scholar 

  2. N. Alon and J. H. Spencer: The probabilistic method, third ed., John Wiley & Sons Inc., Hoboken, NJ, 2008.

    Book  MATH  Google Scholar 

  3. Béla Bollobás: The work of William Timothy Gowers, in: Proceedings of the International Congress of Mathematicians Vol. I (Berlin, 1998), no. Extra Vol. I, 1998, 109–118 (electronic).

    Google Scholar 

  4. C. Borgs, J. T. Chayes, L. Lovász, V. T. Sós and K. Vesztergombi: Convergent sequences of dense graphs. I. Subgraph frequencies, metric properties and testing, Adv. Math. 219 (2008), 1801–1851.

    Article  MathSciNet  MATH  Google Scholar 

  5. D. Conlon and J. Fox: Bounds for graph regularity and removal lemmas, Geom. Funct. Anal. 22 (2012), 1191–1256.

    Article  MathSciNet  MATH  Google Scholar 

  6. J Fox, L. M. Lovász and Y. Zhao: On regularity lemmas and their algorithmic applications, in preparation.

  7. W. T. Gowers: Lower bounds of tower type for Szemerédi’s uniformity lemma, Geom. Funct. Anal. 7 (1997), 322–337.

    Article  MathSciNet  MATH  Google Scholar 

  8. W. Hoeffding: Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 58 (1963), 13–30.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Komlós and M. Simonovits: Szemerédi’s regularity lemma and its applications in graph theory, in: Combinatorics, Paul Erdős is eighty, Vol. 2 (Keszthely, 1993), Bolyai Soc. Math. Stud., vol. 2, János Bolyai Math. Soc., Budapest, 1996, 295–352.

    Google Scholar 

  10. L. Lovász and B. Szegedy: Szemerédi’s lemma for the analyst, Geom. Funct. Anal. 17 (2007), 252–270.

    Article  MathSciNet  MATH  Google Scholar 

  11. G. Moshkovitz and A. Shapira: A short proof of Gowers’ lower bound for the regularity lemma, Combinatorica, to appear.

  12. V. Rödl and M. Schacht: Regularity lemmas for graphs, in: Fete of combinatorics and computer science, Bolyai Soc. Math. Stud., vol. 20, János Bolyai Math. Soc., Budapest, 2010, 287–325.

    Chapter  Google Scholar 

  13. E. Szemerédi: On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975), 199–245.

    Article  MathSciNet  MATH  Google Scholar 

  14. E. Szemerédi: Regular partitions of graphs, in: Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), Colloq. Internat. CNRS, vol. 260, CNRS, Paris, 1978, 399–401.

    Google Scholar 

  15. T. Tao: Szemerédi’s regularity lemma revisited, Contrib. Discrete Math. 1 (2006), 8–28.

    MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    Jacob Fox

  2. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139-4307, USA

    László Miklós Lovász

Authors
  1. Jacob Fox
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  2. László Miklós Lovász
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Corresponding author

Correspondence to László Miklós Lovász.

Additional information

Research supported by a Packard Fellowship, by a Simons Fellowship, by NSF grant DMS-1069197, by an Alfred P. Sloan Fellowship, and by an MIT NEC Corporation Award.

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Fox, J., Lovász, L.M. A tight lower bound for Szemerédi’s regularity lemma. Combinatorica 37, 911–951 (2017). https://doi.org/10.1007/s00493-016-3274-4

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  • DOI: https://doi.org/10.1007/s00493-016-3274-4

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