Abstract
We introduce a correspondence principle (analogous to the Furstenberg Correspondence Principle) that allows one to extract an infinite random graph or hypergraph from a sequence of increasingly large deterministic graphs or hypergraphs. As an application we present a new (infinitary) proof of the hypergraph removal lemma of Nagle-Schacht-Rödl-Skokan and Gowers, which does not require the hypergraph regularity lemma and requires significantly less computation. This in turn gives new proofs of several corollaries of the hypergraph removal lemma, such as Szemerédi’s Theorem on arithmetic progressions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Ajtai and E. Szemerédi, Sets of lattice points that form no squares, Studia Sci. Math. Hungar. 9 (1974), 9–11.
N. Alon and A. Shapira, A characterization of the (natural) graph properties testable with one-sided error, Proc. of FOCS, 2005, pp. 429–438.
F. Chung and R. Graham, Quasi-random hypergraphs, Random Structures Algorithms 1 (1990), 105–124.
P. Erdös, P. Frankl and V. Rödl, The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent, Graphs Combin. 2 (1986), 113–121.
P. Frankl and V. Rödl, The uniformity lemma for hypergraphs, Graphs Combin. 8 (1992), 309–312.
P. Frankl and V. Rödl, Extremal problems on set systems, Random Structures Algorithms 20 (2002), 131–164.
H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math. 31 (1977), 204–256.
H. Furstenberg, Recurrence in Ergodic theory and Combinatorial Number Theory, Princeton University Press, Princeton, NJ, 1981.
H. Furstenberg and Y. Katznelson, An ergodic Szemerédi theorem for commuting transformations, J. Analyse Math. 34 (1978), 275–291.
H. Furstenberg, Y. Katznelson and D. Ornstein, The ergodic theoretical proof of Szemerédi’s theorem, Bull. Amer. Math. Soc. 7 (1982), 527–552.
J-Y. Girard, Herbrand’s theorem and proof theory, Proceedings of the Herbrand Symposium (Marseilles, 1981), North-Holland, Amsterdam, 1982, pp. 29–38.
T. Gowers, Quasirandomness, counting and regularity for 3-uniform hypergraphs, Combin. Probab. Comput. 15 (2006), 143–184.
T. Gowers, Hypergraph regularity and the multidimensional Szemerédi theorem, preprint. See also arXiv:0710.3032
B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Ann. of Math., to appear. See also arXiv:math/0404188
B. Host, Progressions arithmétiques dans les nombres premiers (d’aprés B. Green and T. Tao), Séminaire Bourbaki, Mars 2005, 57eme année, 2004–2005, no. 944.
B. Host and B. Kra, Nonconventional ergodic averages and nilmanifolds, Ann. of Math. (2) 161 (2005), 397–488.
J. Komlós and M. Simonovits, Szemerédi’s regularity lemma and its applications in graph theory, Combinatorics, Paul Erdos is Eighty, Vol. 2 (Keszthely, 1993), János Bolyai Math. Soc., Budapest, 1996, pp. 295–352.
B. Kra, The Green-Tao Theorem on arithmetic progressions in the primes: an ergodic point of view, Bull. Amer. Math. Soc. (N.S.) 43 (2006), 3–23.
L. Lovász and B. Szegedy, Limits of dense graph sequences, J. Combin. Theory Ser. B 96 (2006), 933–957.
B. Nagle, V. Rödl and M. Schacht, The counting lemma for regular k-uniform hypergraphs, Random Structures Algorithms, 28 (2006), 113–179.
V. Rödl and M. Schacht, Regular partitions of hypergraphs: Counting Lemmas, Combin. Probab. Comput. 16 (2007), 887–901.
V. Rödl, M. Schacht, E. Tengan and N. Tokushige, Density theorems and extremal hypergraph problems, Israel J. Math. 152 (2006), 371–380.
V. Rödl and J. Skokan, Regularity lemma for k-uniform hypergraphs, Random Structures Algorithms 25 (2004), 1–42.
V. Rödl and J. Skokan, Applications of the regularity lemma for uniform hypergraphs, Random Structures Algorithms 28 (2006), 180–194.
I. Ruzsa and E. Szemerédi, Triple systems with no six points carrying three triangles, Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol II, North-Holland, Amsterdam-New York, 1978, pp. 939–945.
J. Solymosi, Note on a generalization of Roth’s theorem, Discrete and Computational Geometry, Springer, Berlin, 2003, pp. 825–827.
E. Szemerédi, On sets of integers containing no four elements in arithmetic progression, Acta Math. Acad. Sci. Hungar. 20 (1969), 89–104.
E. Szemerédi, Regular partitions of graphs, Problémes Combinatoires et Théorie des Graphes, (Colloq. Internat. CNRS, Univ. Orsay, 1976), CNRS, Paris, 1978, pp. 399–401.
T. Tao, Szemerédi’s regularity lemma revisited, Contrib. Discrete Math. 1 (2006), 8–28.
T. Tao, A quantitative ergodic theory proof of Szemerédi’s theorem, Electron. J. Combin. 13 (2006) #R 99.
T. Tao, A variant of the hypergraph removal lemma, J. Combin. Theory Ser. A 113 (2006), 1257–1280.
T. Tao, The Gaussian primes contain arbitrarily shaped constellations, J. Analyse Math. 99 (2006), 109–176.
T. Tao, An ergodic transference theorem, unpublished.
P. Varnavides, On certain sets of positive density, J. London Math. Soc. 34 (1959), 358–360.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tao, T. A correspondence principle between (hyper)graph theory and probability theory, and the (hyper)graph removal lemma. J Anal Math 103, 1–45 (2007). https://doi.org/10.1007/s11854-008-0001-0
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s11854-008-0001-0