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Hereditary quasirandomness without regularity

Published online by Cambridge University Press:  26 January 2017

DAVID CONLON
Affiliation:
Mathematical Institute, Oxford OX2 6GG, United Kingdom. e-mail: david.conlon@maths.ox.ac.uk
JACOB FOX
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, U.S.A. e-mail: jacobfox@stanford.edu
BENNY SUDAKOV
Affiliation:
Department of Mathematics, ETH, 8092 Zurich, Switzerland. e-mail: benjamin.sudakov@math.ethz.ch

Abstract

A result of Simonovits and Sós states that for any fixed graph H and any ε > 0 there exists δ > 0 such that if G is an n-vertex graph with the property that every SV(G) contains pe(H) |S|v(H) ± δ nv(H) labelled copies of H, then G is quasirandom in the sense that every SV(G) contains $\frac{1}{2}$p|S|2± ε n2 edges. The original proof of this result makes heavy use of the regularity lemma, resulting in a bound on δ−1 which is a tower of twos of height polynomial in ε−1. We give an alternative proof of this theorem which avoids the regularity lemma and shows that δ may be taken to be linear in ε when H is a clique and polynomial in ε for general H. This answers a problem raised by Simonovits and Sós.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2017 

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References

REFERENCES

[1] Chung, F. R. K., Graham, R. L. and Wilson, R. M. Quasi-random graphs. Combinatorica 9 (1989), 345362.Google Scholar
[2] Conlon, D. A new upper bound for diagonal Ramsey numbers. Ann. of Math. 170 (2009), 941960.CrossRefGoogle Scholar
[3] Conlon, D. and Fox, J. Bounds for graph regularity and removal lemmas. Geom. Funct. Anal. 22 (2012), 11911256.Google Scholar
[4] Conlon, D., Fox, J. and Sudakov, B. An approximate version of Sidorenko's conjecture. Geom. Funct. Anal. 20 (2010), 13541366.Google Scholar
[5] Conlon, D., Kim, J. H., Lee, C. and Lee, J. Some advances on Sidorenko's conjecture. arXiv:1510.06533 [math.CO].Google Scholar
[6] Erdős, P., Goldberg, M., Pach, J. and Spencer, J. Cutting a graph into two dissimilar halves. J. Graph Theory 12 (1988), 121131.Google Scholar
[7] Gowers, W. T. A new proof of Szemerédi's theorem. Geom. Funct. Anal. 11 (2001), 465588.CrossRefGoogle Scholar
[8] Hoory, S., Linial, N. and Wigderson, A. Expander graphs and their applications. Bull. Amer. Math. Soc. 43 (2006), 439561.Google Scholar
[9] Janson, S., Łuczak, T. and Ruciński, A. Random graphs. (Wiley, New York, 2000).Google Scholar
[10] Katona, G. A theorem of finite sets. Theory of graphs (Proc. Colloq., Tihany, 1966) (Academic Press, New York, 1968), 187207.Google Scholar
[11] Kim, J. H., Lee, C. and Lee, J. Two approaches to Sidorenko's conjecture. Trans. Amer. Math. Soc. 368 (2016), 50575074.CrossRefGoogle Scholar
[12] Krivelevich, M. and Sudakov, B. Pseudo-random graphs. More sets, graphs and numbers. Bolyai Soc. Math. Stud. 15 (Springer, Berlin, 2006), 199262.Google Scholar
[13] Kruskal, J. B. The number of simplices in a complex. Mathematical optimization techniques, (Univ. of California Press, Berkeley, Calif., 1963), 251278.Google Scholar
[14] Li, J. X. and Szegedy, B. On the logarithmic calculus and Sidorenko's conjecture. To appear in Combinatorica.Google Scholar
[15] Lovász, L. Combinatorial Problems and Exercises, 2nd edition (AMS Chelsea Publishing, Providence, RI, 2007).Google Scholar
[16] Lovász, L. Large networks and graph limits. Amer. Math. Soc. Colloq. Publ., vol. 60 (Amer. Math. Soc., Providence, RI, 2012).CrossRefGoogle Scholar
[17] Lovász, L. and Szegedy, B. Szemerédi's lemma for the analyst. Geom. Funct. Anal. 17 (2007), 252270.Google Scholar
[18] Reiher, C. and Schacht, M. in preparation.Google Scholar
[19] Shapira, A. Quasi-randomness and the distribution of copies of a fixed graph. Combinatorica 28 (2008), 735745.Google Scholar
[20] Simonovits, M. and Sós, V. T. Hereditarily extended properties, quasi-random graphs and not necessarily induced subgraphs. Combinatorica 17 (1997), 577596.Google Scholar
[21] Skokan, J. and Thoma, L. Bipartite subgraphs and quasi-randomness. Graphs Combin. 20 (2004), 255262.Google Scholar
[22] Szegedy, B. An information theoretic approach to Sidorenko's conjecture. arXiv:1406.6738v3 [math.CO].Google Scholar
[23] Szemerédi, E. Regular partitions of graphs. Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), 399–401. Colloq. Internat. CNRS, vol. 260 (CNRS, Paris, 1978).Google Scholar
[24] Thomason, A. Pseudorandom graphs. Random graphs '85 (Poznań, 1985) North-Holland Math. Stud. 144 (North-Holland, Amsterdam, 1987), 307331.Google Scholar
[25] Thomason, A. Random graphs, strongly regular graphs and pseudorandom graphs. Surveys in Combinatorics 1987 (New Cross, 1987), 173–195. London Math. Soc. Lecture Note Ser. 123 (Cambridge University Press, Cambridge, 1987).Google Scholar