Book contents
- Frontmatter
- Contents
- Preface
- 1 Graph removal lemmas
- 2 The geometry of covering codes: small complete caps and saturating sets in Galois spaces
- 3 Bent functions and their connections to combinatorics
- 4 The complexity of change
- 5 How symmetric can maps on surfaces be?
- 6 Some open problems on permutation patterns
- 7 The world of hereditary graph classes viewed through Truemper configurations
- 8 Structure in minor-closed classes of matroids
- 9 Automatic counting of tilings of skinny plane regions
- References
1 - Graph removal lemmas
Published online by Cambridge University Press: 05 July 2013
Book contents
- Frontmatter
- Contents
- Preface
- 1 Graph removal lemmas
- 2 The geometry of covering codes: small complete caps and saturating sets in Galois spaces
- 3 Bent functions and their connections to combinatorics
- 4 The complexity of change
- 5 How symmetric can maps on surfaces be?
- 6 Some open problems on permutation patterns
- 7 The world of hereditary graph classes viewed through Truemper configurations
- 8 Structure in minor-closed classes of matroids
- 9 Automatic counting of tilings of skinny plane regions
- References
Summary
Abstract
The graph removal lemma states that any graph on n vertices with o(nh) copies of a fixed graph H on h vertices may be made H-free by removing o(n2) edges. Despite its innocent appearance, this lemma and its extensions have several important consequences in number theory, discrete geometry, graph theory and computer science. In this survey we discuss these lemmas, focusing in particular on recent improvements to their quantitative aspects.
Introduction
The triangle removal lemma states that for every ε > 0 there exists δ > 0 such that any graph on n vertices with at most δn3 triangles may be made triangle-free by removing at most εn2 edges. This result, proved by Ruzsa and Szemerédi [94] in 1976, was originally stated in rather different language.
The original formulation was in terms of the (6, 3)-problem. This asks for the maximum number of edges f(3)(n, 6, 3) in a 3-uniform hypergraph on n vertices such that no 6 vertices contain 3 edges. Answering a question of Brown, Erdὄs and Sós [19], Ruzsa and Szemerédi showed that f(3)(n, 6, 3) = o(n2). Their proof used several iterations of an early version of Szemerédi's regularity lemma [111].
This result, developed by Szemerédi in his proof of the Erdὄos-Turán conjecture on arithmetic progressions in dense sets [110], states that every graph may be partitioned into a small number of vertex sets so that the graph between almost every pair of vertex sets is random-like.
- Type
- Chapter
- Information
- Surveys in Combinatorics 2013 , pp. 1 - 50Publisher: Cambridge University PressPrint publication year: 2013
References
[1] and , Sets of lattice points that form no squares, Stud. Sci. Math. Hungar. 9 (1974), 9–11.Google Scholar
[2] , Explicit Ramsey graphs and orthonormal labellings, Elec¬tron. J. Combin. 1 (1994), R12, 8pp.Google Scholar
[3] , Testing subgraphs in large graphs, Random Structures Al¬gorithms 21 (2002), 359–370.Google Scholar
[4] , , , and , The algorithmic aspects of the regularity lemma, J. Algorithms 16 (1994), 80–109.Google Scholar
[5] , , and , Random sampling and approximation of MAX-CSPs, J. Comput. System Sci. 67 (2003), 212–243.Google Scholar
[7] and , Testing perfectness is hard, submitted.
[8] , and , Nearly complete graphs decomposable into large induced matchings and their applications, in Proc. of STOC 2012, 1079–1090, to appear in J. European Math. Soc.
[9] and , Testing subgraphs in directed graphs, J. Comput. System Sci. 69 (2004), 353–382.Google Scholar
[10] and , A characterization ofeasily testable induced subgraphs, Combin. Probab. Comput. 15 (2006), 791–805.Google Scholar
[11] and , Every monotone graph property is testable, in Proc. of STOC 2005, 128–137, SIAM J. Comput. (Special Issue on STOC '05) 38 (2008), 505–522.Google Scholar
[12] and , A characterization of the (natural) graph properties testable with one-sided error, in Proc. of FOCS 2005, 429–438, SIAM J. Comput. (Special Issue on FOCS '05) 37 (2008), 1703–1727.Google Scholar
[13] and , Testability and repair ofhereditary hypergraph properties, Random Structures Algorithms 36 (2010), 373–463.Google Scholar
[14] , and , Every monotone 3-graph property is testable, SIAM J. Discrete Math. 21 (2007), 73–92.Google Scholar
[15] , and , Independent sets in hyper-graphs, submitted.
[16] , On sets of integers which contain no three terms in arithmetic progression, Proc. Nat. Acad. Sci. 32 (1946), 331–332.Google Scholar
[17] , Random graphs, second edition, Cambridge Studies in Advanced Mathematics 73, Cambridge University Press, Cambridge, 2001.
[18] , , , and , Convergent sequences of dense graphs I: subgraph frequencies, metric properties and testing, Adv. Math. 219 (2008), 1801–1851.Google Scholar
[19] , and , On the existence of triangulated spheres in 3-graphs, and related problems, Period. Math. Hungar. 3 (1973), 221–228.Google Scholar
[20] , Developments at the interface between combinatorics and Fourier analysis, PhD thesis, University of Cambridge, 2009.
[23] , , and , Extremal problems for local properties of graphs, in Combinatorial mathematics and combinatorial computing (Palmerston North, 1990), Australas. J. Combin. 4 (1991), 25–31.Google Scholar
[24] and , Bounds for graph regularity and removal lemmas, Geom. Funct. Anal. 22 (2012), 1192–1256.Google Scholar
[25] and , Combinatorial theorems in sparse random sets, submitted.
[26] , , and , On the KŁR conjecture in random graphs, submitted.
[27] , and , Extremal results in sparse pseudorandom graphs, submitted.
[28] , and , A fast approximation algorithm for computing the frequencies of subgraphs in a given graph, SIAM J. Comput. 24 (1995), 598–620.Google Scholar
[29] , A lower bound for the largest number of triangles with a common edge, 1977, unpublished manuscript.
[30] , Some problems on finite and infinite graphs, in Logic and combinatorics (Arcata, Calif., 1985), 223–228, Contemp. Math. 65, Amer. Math. Soc., Providence, RI, 1987.
[31] , Some of my favourite problems in various branches of combinatorics, in Combinatorics 92 (Catania, 1992), Matematiche (Catania) 47 (1992), 231–240.Google Scholar
[32] , and , The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent, Graphs Combin. 2 (1986), 113–121.Google Scholar
[34] and , On the evolution of random graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 (1960), 17–61.Google Scholar
[35] and , A limit theorem in graph theory, Studia Sci. Math. Hungar. 1 (1966), 51–57.Google Scholar
[36] and , On the structure of linear graphs, Bull. Amer.Math.Soc. 52 (1946), 1087–1091.Google Scholar
[37] , , , and , Monotonicity testing over general poset domains, in Proceedings of the 2002 ACM Symposium on Theory of Computing, 474–483, ACM, New York, 2002.
[39] and , On a problem of Erdős and Rothschild on edges in triangles, to appear in Combinatorica.
[40] and , Exact solution of some Turán-type problems, J. Combin. Theory Ser. A 45 (1987), 226–262.Google Scholar
[41] , and , On subsets of abelian groups with no 3-term arithmetic progression, J. Combin. Theory Ser. A 45 (1987), 157–161.Google Scholar
[42] and , Extremal problems on set systems, Random Structures Algorithms 20 (2002), 131–164.Google Scholar
[43] , and , Ramsey properties of discrete random structures, Random Structures Algorithms, 37 (2010), 407–436.Google Scholar
[44] and , The regularity lemma and approximation schemes for dense problems, in Proceedings of the 37th IEEE FOCS (1996), 12–20.
[45] and , Quick approximation to matrices and applications, Combinatorica 19 (1999), 175–220.Google Scholar
[46] , The maximum number of edges in a minimal graph of diameter 2, J. Graph Theory 16 (1992), 81–98.Google Scholar
[47] , Extremal hypergraphs and combinatorial geometry, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 1343–1352, Birkhäuser, Basel, 1995.
[48] , Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math. 31 (1977), 204–256.Google Scholar
[49] and , An ergodic Szemerédi theorem for commuting transformations, J. Analyse Math. 34 (1978), 275–291.Google Scholar
[50] , and , The ergodic theoretical proof of Szemerédi's theorem, Bull. Amer. Math. Soc. 7 (1982), 527–552.Google Scholar
[51] and , The sparse regularity lemma and its applications, in Surveys in Combinatorics 2005, 227–258, London Math. Soc. Lecture Note Ser. 327, Cambridge Univ. Press, Cambridge, 2005.
[52] , and , Property testing and its applications to learning and approximation, J. ACM 45 (1998), 653–750.Google Scholar
[53] , Lower bounds of tower type for Szemerédi's uniformity lemma, Geom. Funct. Anal. 7 (1997), 322–337.Google Scholar
[54] , A new proof of Szemerédi's theorem for arithmetic progressions of length four, Geom. Funct. Anal. 8 (1998), 529–551.Google Scholar
[56] , Quasirandomness, counting and regularity for 3-uniform hypergraphs, Combin. Probab. Comput. 15 (2006), 143–184.Google Scholar
[57] , Hypergraph regularity and the multidimensional Szemerédi theorem, Ann. of Math. 166 (2007), 897–946.Google Scholar
[58] , A Szemerédi-type regularity lemma in abelian groups, with applications, Geom. Funct. Anal. 15 (2005), 340–376.Google Scholar
[59] , and , Turán's extremal problem in random graphs: forbidding odd cycles, Combinatorica 16 (1996), 107–122.Google Scholar
[60] , A simple regularization of hypergraphs, submitted.
[61] , and , Random graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience, New York, 2000.
[62] and , A wowzer-type lower bound for the strong regularity lemma, Proc. London Math. Soc., to appear.
[63] and , Solution of a problem of P. Erdős about the maximum number of triangles with a common edge in a graph, C. R. Acad. Bulgare Sci. 32 (1979), 1315–1318.Google Scholar
[64] , Szemerédi's regularity lemma for sparse graphs, in Foundations of computational mathematics (Rio de Janeiro, 1997), Springer, Berlin, 1997, 216–230.
[65] , and , Arithmetic progressions of length three in subsets of a random set, Acta Arith. 75 (1996), 133–163.Google Scholar
[67] , , and , Weak regularity and linear hypergraphs, J. Combin. Theory Ser. B 100 (2010), 151–160.Google Scholar
[68] , , , and , The hypergraph regularity method and its applications, Proc. Natl. Acad. Sci. USA 102 (2005), 8109–8113.Google Scholar
[69] , , and , On the triangle removal lemma for subgraphs of sparse pseudorandom graphs, in An Irregular Mind (Szemerédi is 70), Bolyai Society Math. Studies 21, Springer, 2010, 359–404.
[70] , and , A removal lemma for linear systems over finite fields, in Sixth conference on discrete mathematics and computer science (Spanish), 417–423, Univ. Lleida, Lleida, 2008.
[71] , and , A combinatorial proof of the removal lemma for groups, J. Combin. Theory Ser. A 116 (2009), 971–978.Google Scholar
[72] , and , A removal lemma for systems of linear equations over finite fields, Israel J. Math. 187 (2012), 193–207.Google Scholar
[73] , and , On the removal lemma for linear systems over abelian groups, European J. Combin. 34 (2013), 248–259.Google Scholar
[74] and , Pseudo-random graphs, in More sets, graphs and numbers, Bolyai Soc. Math. Stud. 15, Springer, Berlin, 2006, 199–262.
[76] and , Testing properties of graphs and functions, Israel J. Math. 178 (2010), 113–156.Google Scholar
[77] , Randomness and regularity, in International Congress of Mathematicians, Vol. III, 899–909, Eur. Math. Soc., Zürich, 2006.
[78] and , Regularity properties for triple systems, Random Structures Algorithms 23 (2003), 264–332.Google Scholar
[79] , and , The counting lemma for regular k-uniform hypergraphs, Random Structures Algorithms 28 (2006), 113–179.Google Scholar
[81] , and , Counting small cliques in 3-uniform hypergraphs, Combin. Probab. Comput. 14 (2005), 371–413.Google Scholar
[82] , A new proof of the density Hales-Jewett theorem, Ann. of Math. 175 (2012), 1283–1327.Google Scholar
[84] and , Lower bounds on probability thresholds for Ramsey properties, in Combinatorics, Paul Erdőos is eighty, Vol. 1, 317–346, Bolyai Soc. Math. Stud., János Bolyai Math. Soc., Budapest, 1993.
[85] and , Threshold functions for Ramsey properties, J. Amer. Math. Soc. 8 (1995), 917–942.Google Scholar
[86] and , Regular partitions of hypergraphs: regularity lemmas, Combin. Probab. Comput. 16 (2007), 833–885.Google Scholar
[88] and , Regularity lemmas for graphs, in Fete of Combinatorics and Computer Science, Bolyai Soc. Math. Stud. 20, Springer, 2010, 287–325.
[89] and , Regularity lemma for uniform hypergraphs, Random Structures Algorithms 25 (2004), 1–42.Google Scholar
[90] and , Counting subgraphs in quasi-random 4-uniform hypergraphs, Random Structures Algorithms 26 (2005), 160–203.Google Scholar
[91] and , Applications of the regularity lemma for uniform hypergraphs, Random Structures Algorithms 28 (2006), 180–194.Google Scholar
[93] and , Robust characterization of polynomials with applications to program testing, SIAM J. Comput. 25 (1996), 252–271.Google Scholar
[94] and , Triple systems with no six points carrying three triangles, in Combinatorics (Keszthely, 1976), Coll. Math. Soc. J. Bolyai 18, Volume II, 939–945.Google Scholar
[95] , Stability results for random discrete structures, to appear in Random Structures Algorithms.
[98] and , Hypergraph containers, submitted.
[99] , Extremal results for random discrete structures, submitted.
[100] and , Roth's theorem in many variables, submitted.
[101] , Green's conjecture and testing linear-invariant properties, in Proceedings of the 2009 ACM International Symposium on Theory of Computing, 159–166, ACM, New York, 2009.
[102] , A proof of Green's conjecture regarding the removal properties of sets of linear equations, J. London Math. Soc. 81 (2010), 355–373.Google Scholar
[103] , On a generalization of Szemerédi's theorem, Proc. London Math. Soc. 93 (2006), 723–760.Google Scholar
[104] , A method for solving extremal problems in graph theory, stability problems, in Theory of graphs (Proc. Colloq. Tihany, 1966), Academic Press, New York, 1968, 279–319.
[105] , Note on a generalization of Roth's theorem, in Discrete and computational geometry, Algorithms Combin. Vol. 25, Springer, 2003, 825–827.
[106] , A note on a question of Erdős and Graham, Combin. Probab. Comput. 13 (2004), 263–267.Google Scholar
[107] , Regularity, uniformity, and quasirandomness, Proc. Natl. Acad. Sci. USA 102 (2005), 8075–8076.Google Scholar
[108] , and , A generalization of Turán's theorem, J. Graph Theory 49 (2005), 187–195.Google Scholar
[109] , The symmetry preserving removal lemma, Proc. Amer. Math. Soc. 138 (2010), 405–408.Google Scholar
[110] , Integer sets containing no k elements in arithmetic progression, Acta Arith. 27 (1975), 299–345.Google Scholar
[111] , Regular partitions of graphs, in Colloques Internationaux CNRS 260 – Problèmes Combinatoires et Théorie des Graphes, Orsay, (1976), 399–401.
[112] , Szemerédi's regularity lemma revisited, Contrib. Discrete Math. 1 (2006), 8–28.Google Scholar
[113] , A variant of the hypergraph removal lemma, J. Combin. Theory Ser. A 113 (2006), 1257–1280.Google Scholar
[114] , Pseudorandom graphs, in Random graphs '85 (Poznan, 1985), 307–331, North-Holland Math. Stud. 144, North-Holland, Amsterdam, 1987.
[115] , 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 Univ. Press, Cambridge, 1987.
[116] , Eine Extremalaufgabe aus der Graphentheorie, Mat. Fiz. Lapok 48 (1941), 436–452.Google Scholar
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