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Summer School on Theoretical Aspects of Computer Science

TACSci 2000: Theoretical Aspects of Computer Science pp 84–112Cite as

The Regularity Lemma and Its Applications in Graph Theory

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2292))

Abstract

Szemerédi’s Regularity Lemma is an important tool in discrete mathematics. It says that, in some sense, all graphs can be approximated by random-looking graphs. Therefore the lemma helps in proving theorems for arbitrary graphs whenever the corresponding result is easy for random graphs. In the last few years more and more new results were obtained by using the Regularity Lemma, and also some new variants and generalizations appeared. Komlós and Simonovits have written a survey on the topic [96]. The present survey is, in a sense, a continuation of the earlier survey. Here we describe some sample applications and generalizations. To keep the paper self-contained we decided to repeat (sometimes in a shortened form) parts of the first survey, but the emphasis is on new results.

Research supported in part by the NSF grant DMS-9801396.

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References

  1. M. Ajtai, J. Komlós, M. Simonovits and E. Szemerédi, Solution of the Erdős-Sós Conjecture, in preparation.

    Google Scholar 

  2. N. Alon, R. Duke, H. Leffman, V. Rödl, R. Yuster, The algorithmic aspects of the regularity lemma, FOCS 33 (1992), 479-481, Journal of Algorithms 16 (1994), 80–109.

    Article  MATH  MathSciNet  Google Scholar 

  3. N. Alon, E. Fischer, 2-factors in dense graphs, Discrete Math.

    Google Scholar 

  4. N. Alon, R. Yuster, Almost H-factors in dense graphs, Graphs and Combinatorics 8 (1992), 95–102.

    Article  MATH  MathSciNet  Google Scholar 

  5. N. Alon, R. Yuster, H-factors in dense graphs, J. Combinatorial Theory B66 (1996), 269–282.

    MathSciNet  Google Scholar 

  6. L. Babai, M. Simonovits, J. Spencer, Extremal subgraphs of random graphs, Journal of Graph Theory 14 (1990), 599–622.

    Article  MATH  MathSciNet  Google Scholar 

  7. V. Bergelson, A. Leibman, Polynomial extension of van der Waerden’s and Szemerédi’s theorem.

    Google Scholar 

  8. B. Bollobás, Extremal graph theory, Academic Press, London (1978).

    MATH  Google Scholar 

  9. Béla Bollobás, The work of William Timothy Gowers, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), Doc. Math. 1998, Extra Vol. I, 109-118 (electronic).

    Google Scholar 

  10. B. Bollobás, P. Erdős, On a Ramsey-Turán type problem, Journal of Combinatorial Theory B21 (1976), 166–168.

    Article  Google Scholar 

  11. B. Bollobás, P. Erdős, M. Simonovits, E. Szemerédi, Extremal graphs without large forbidden subgraphs, Annals of Discrete Mathematics 3 (1978), 29–41, North-Holland.

    Article  MATH  MathSciNet  Google Scholar 

  12. B. Bollobás, A. Thomason, The structure of hereditary properties and colourings of random graphs. Combinatorica 20 (2000), 173–202.

    Article  MATH  MathSciNet  Google Scholar 

  13. W. G. Brown, P. Erdős, M. Simonovits, Extremal problems for directed graphs, Journal of Combinatorial Theory B15 (1973), 77–93.

    Article  Google Scholar 

  14. W. G. Brown, P. Erdős, M. Simonovits, Inverse extremal digraph problems, Colloq. Math. Soc. J. Bolyai 37 (Finite and Infinite Sets), Eger (Hungary) 1981, Akad. Kiadó, Budapest (1985), 119–156.

    Google Scholar 

  15. W. G. Brown, P. Erdős, M. Simonovits, Algorithmic solution of extremal digraph problems, Transactions of the American Math. Soc. 292/2 (1985), 421–449.

    Article  Google Scholar 

  16. W. G. Brown, P. Erdős, V. T. Sós, Some extremal problems on r-graphs, New directions in the theory of graphs (Proc. Third Ann Arbor Conf., Univ. Michigan, Ann Arbor, Mich, 1971), 53–63. Academic Press, New York, 1973.

    Google Scholar 

  17. W. G. Brown, P. Erdős, V. T. Sós, On the existence of triangulated spheres in 3-graphs, and related problems, Period. Math. Hungar. 3 (1973), 221–228.

    Article  MATH  MathSciNet  Google Scholar 

  18. W. G. Brown, M. Simonovits, Digraph extremal problems, hypergraph extremal problems, and densities of graph structures, Discrete Mathematics 48 (1984), 147–162.

    Article  MATH  MathSciNet  Google Scholar 

  19. S. Burr, P. Erdős, P. Frankl, R. L. Graham, V. T. Sós, Further results on maximal antiramsey graphs, Proc. Kalamazoo Combin. Conf. (1989), 193–206.

    Google Scholar 

  20. S. Burr, P. Erdős, R. L. Graham, V. T. Sós, Maximal antiramsey graphs and the strong chromatic number (The nonbipartite case) Journal of Graph Theory 13 (1989), 163–182.

    Article  Google Scholar 

  21. L. Caccetta, R. Häggkvist, On diameter critical graphs, Discrete Mathematics 28 (1979), 223–229.

    Article  MATH  MathSciNet  Google Scholar 

  22. Fan R. K. Chung, Regularity lemmas for hypergraphs and quasi-randomness, Random Structures and Algorithms 2 (1991), 241–252.

    Article  MATH  MathSciNet  Google Scholar 

  23. F. R. K. Chung, R. L. Graham, R. M. Wilson, Quasi-random graphs, Combinatorica 9 (1989), 345–362.

    Article  MATH  MathSciNet  Google Scholar 

  24. V. Chvátal, V. Rödl, E. Szemerédi, W. T. Trotter Jr., The Ramsey number of a graph with bounded maximum degree, Journal of Combinatorial Theory B34 (1983), 239–243.

    Article  Google Scholar 

  25. V. Chvátal, E. Szemerédi, On the Erdős-Stone theorem, Journal of the London Math. Soc. 23 (1981), 207–214.

    Article  MATH  Google Scholar 

  26. V. Chvátal, E. Szemerédi, Notes on the Erdős-Stone theorem, Combinatorial Mathematics, Annals of Discrete Mathematics 17 (1983), (Marseille-Luminy, 1981), 183–190, North-Holland, Amsterdam-New York, 1983.

    MATH  Google Scholar 

  27. K. Corrádi, A. Hajnal, On the maximal number of independent circuits in a graph, Acta Math. Acad. Sci. Hung. 14 (1963), 423–439.

    Article  MATH  Google Scholar 

  28. W. Deuber, Generalizations of Ramsey’s theorem, Proc. Colloq. Math. Soc. János Bolyai 10 (1974), 323–332.

    MathSciNet  Google Scholar 

  29. G. A. Dirac, Some theorems on abstract graphs, Proc. London Math. Soc. 2 (1952), 68–81.

    Article  MathSciNet  Google Scholar 

  30. Duke, Richard A., Hanno Lefmann, Hanno, Vojtěch Rödl, A fast approximation algorithm for computing the frequencies of subgraphs in a given graph, SIAM J. Comput. 24 (1995), 598–620.

    Article  MATH  MathSciNet  Google Scholar 

  31. R. A. Duke, V. Rödl, On graphs with small subgraphs of large chromatic number, Graphs Combin. 1 (1985), 91–96.

    Article  MATH  MathSciNet  Google Scholar 

  32. R. A. Duke, V. Rödl, The Erdös-Ko-Rado theorem for small families, J. Combin. Theory Ser. A65 (1994), 246–251.

    Article  Google Scholar 

  33. P. Erdős, Some recent results on extremal problems in graph theory, Results, International Symposium, Rome (1966), 118–123.

    Google Scholar 

  34. P. Erdős, On some new inequalities concerning extremal properties of graphs, Theory of Graphs, Proc. Coll. Tihany, Hungary (P. Erdős and G. Katona eds.) Acad. Press N. Y. (1968), 77–81.

    Google Scholar 

  35. P. Erdős, On some extremal problems on r-graphs, Discrete Mathematics 1 (1971), 1–6.

    Article  MathSciNet  Google Scholar 

  36. P. Erdős, Some old and new problems in various branches of combinatorics, Proc. 10th Southeastern Conf. on Combinatorics, Graph Theory and Computation, Boca Raton (1979) Vol I., Congressus Numerantium 23 (1979), 19–37.

    Google Scholar 

  37. P. Erdős, On the combinatorial problems which I would most like to see solved, Combinatorica 1 (1981), 25–42.

    Article  MathSciNet  Google Scholar 

  38. P. Erdős, P. Frankl, V. Rödl, The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent, Graphs and Combinatorics 2 (1986), 113–121.

    Article  MathSciNet  Google Scholar 

  39. P. Erdős, Z. Füredi, M. Loebl, V. T. Sós, Studia Sci. Math. Hung. 30 (1995), 47–57. (Identical with the book Combinatorics and its applications to regularity and irregularity of structures, W. A. Deuber and V. T. Sós eds., Akadémiai Kiadó, 47-58.)

    Google Scholar 

  40. P. Erdős, T. Gallai, On maximal paths and circuits of graphs, Acta Math. Acad. Sci. Hung. 10 (1959), 337–356.

    Article  Google Scholar 

  41. P. Erdős, A. Hajnal, On complete topological subgraphs of certain graphs, Annales Univ. Sci. Budapest 7 (1969), 193–199.

    Google Scholar 

  42. P. Erdős, A. Hajnal, L. Pósa, Strong embedding of graphs into colored graphs, Proc. Colloq. Math. Soc. János Bolyai 10 (1975), 585–595.

    Google Scholar 

  43. P. Erdős, A. Hajnal, M. Simonovits, V. T. Sós, E. Szemerédi, Turán-Ramsey theorems and simple asymptotically extremal structures, Combinatorica 13 (1993), 31–56.

    Article  MathSciNet  Google Scholar 

  44. P. Erdős, A. Hajnal, M. Simonovits, V. T. Sós, E. Szemerédi, Turán-Ramsey theorems for Kp-stability numbers, Proc. Cambridge, also in Combinatorics, Probability and Computing 3 (1994) (P. Erdős birthday meeting), 297–325.

    Google Scholar 

  45. P. Erdős, A. Hajnal, V. T. Sós, E. Szemerédi, More results on Ramsey-Turán type problems, Combinatorica 3 (1983), 69–81.

    Article  MathSciNet  Google Scholar 

  46. P. Erdős, M. Simonovits, A limit theorem in graph theory, Studia Sci. Math. Hung. 1 (1966), 51–57.

    Google Scholar 

  47. P. Erdős, M. Simonovits, The chromatic properties of geometric graphs, Ars Combinatoria 9 (1980), 229–246.

    MathSciNet  Google Scholar 

  48. P. Erdős, M. Simonovits, Supersaturated graphs and hypergraphs, Combinatorica 3 (1983), 181–192.

    Article  MathSciNet  Google Scholar 

  49. P. Erdős, M. Simonovits, How many colours are needed to colour every pentagon of a graph in five colours? (to be published).

    Google Scholar 

  50. P. Erdős, V. T. Sós, The tree conjecture, Mentioned in P. Erdős, Extremal problems in graph theory, Theory of graphs and its applications, Proc. of the Symposium held in Smolenice in June 1963, 29–38.

    Google Scholar 

  51. P. Erdős, V. T. Sós, Some remarks on Ramsey’s and Turán’s theorem, Combinatorial theory and its applications, II (Proc. Colloq., Balatonfüred, 1969), 395–404, North-Holland, Amsterdam, 1970.

    Google Scholar 

  52. P. Erdős, A. H. Stone, On the structure of linear graphs, Bull. Amer. Math. Soc. 52 (1946), 1089–1091.

    Google Scholar 

  53. P. Erdős, P. Turán, On some sequences of integers, J. London Math. Soc. 11 (1936), 261–264.

    Article  Google Scholar 

  54. G. Fan, R. Häggkvist, The square of a hamiltonian cycle, SIAM J. Disc. Math.

    Google Scholar 

  55. G. Fan, H. A. Kierstead, The square of paths and cycles, Journal of Combinatorial Theory B63 (1995), 55–64.

    Article  MathSciNet  Google Scholar 

  56. G. Fan, H. A. Kierstead, The square of paths and cycles II.

    Google Scholar 

  57. R. J. Faudree, R. J. Gould, M. Jacobson, On a problem of Pósa and Seymour.

    Google Scholar 

  58. R. J. Faudree, R. J. Gould, M. S. Jacobson, R. H. Schelp, Seymour’s conjecture, Advances in Graph Theory (V. R. Kulli ed.), Vishwa International Publications (1991), 163–171.

    Google Scholar 

  59. P. Frankl, Z. Füredi, Exact solution of some Turán-type problems, Journal of Combinatorial Theory A45 (1987), 226–262.

    Google Scholar 

  60. P. Frankl, J. Pach, An extremal problem on Kr-free graphs, Journal of Graph Theory 12 (1988), 519–523.

    Article  MATH  MathSciNet  Google Scholar 

  61. P. Frankl, V. Rödl, The Uniformity Lemma for hypergraphs, Graphs and Combinatorics 8 (1992), 309–312.

    Article  MATH  MathSciNet  Google Scholar 

  62. Alan Frieze, Ravi Kannan, Quick approximation to matrices and applications, Combinatorica 19 (1999), 175–220.

    Article  MATH  MathSciNet  Google Scholar 

  63. Alan Frieze, Ravi Kannan, A simple algorithm for constructing Szemerédi’s regularity partition, Electron. J. Combin. 6 (1999), Research Paper 17 (electronic).

    Google Scholar 

  64. Z. Füredi, Turán type problems, in Surveys in Combinatorics (1991), Proc. of the 13th British Combinatorial Conference, (A. D. Keedwell ed.) Cambridge Univ. Press. London Math. Soc. Lecture Note Series 166 (1991), 253–300.

    Google Scholar 

  65. Z. Füredi, The maximum number of edges in a minimal graph of diameter 2, Journal of Graph Theory 16 (1992), 81–98.

    Article  MATH  MathSciNet  Google Scholar 

  66. H. Fcurstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, Journal d’Analyse Math. 31 (1977), 204–256.

    Article  Google Scholar 

  67. H. Fürstenberg, A polynomial Szemerédi theorem, Combinatorics, Paul Erdős is eighty, Vol. 2 (Keszthely, 1993), 1–16, Bolyai Soc. Math. Stud., 2, János Bolyai Math. Soc., Budapest, 1996

    Google Scholar 

  68. H. Fürstenberg, Y. Katznelson, Idempotents in compact semigroups and Ramsey theory, Israel Journal of Mathematics 68 (1989), 257–270.

    Article  MathSciNet  Google Scholar 

  69. H. Fürstenberg, Y. Katznelson, A density version of the Hales-Jewett theorem, Journal d'Analyse Math. 57 (1991), 64–119.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  71. R. L. Graham, B. L. Rothschild, J. Spencer, Ramsey Theory, Wiley Interscience, Series in Discrete Mathematics (1980).

    Google Scholar 

  72. A. Hajnal, W. Maass, Gy. Turán, On the communication complexity of graph properties, 20th STOC, Chicago (1988), 186–191.

    Google Scholar 

  73. A. Hajnal, E. Szemerédi, Proof of a conjecture of Erdős, Combinatorial Theory and its Applications vol. II (P. Erdős, A. Rényi and V. T. Sós eds.), Colloq. Math. Soc. J. Bolyai 4, North-Holland, Amsterdam (1970), 601–623.

    Google Scholar 

  74. P. E. Haxell, Y. Kohayakawa, The size-Ramsey number of trees, Israel J. Math. 89 (1995), 261–274.

    Article  MATH  MathSciNet  Google Scholar 

  75. P. E. Haxell, Y. Kohayakawa, On an anti-Ramsey property of Ramanujan graphs, Random Structures and Algorithms 6 (1995), 417–431.

    Article  MATH  MathSciNet  Google Scholar 

  76. P. E. Haxell, Y. Kohayakawa, T. Luczak, The induced size-Ramsey number of cycles, Combinatorics, Probability and Computing 4 (1995), 217–239.

    Article  MATH  MathSciNet  Google Scholar 

  77. P. E. Haxell, Y. Kohayakawa, T. Luczak, Turán’s extremal problem in random graphs: forbidding even cycles, Journal of Combinatorial Theory B64 (1995), 273–287.

    Article  MathSciNet  Google Scholar 

  78. P. E. Haxell, Y. Kohayakawa, T. Luczak, Turán’s extremal problem in random graphs: forbidding odd cycles, Combinatorica 16 (1996), 107–122.

    Article  MATH  MathSciNet  Google Scholar 

  79. P. E. Haxell, T. Luczak, P. W. Tingley, Ramsey Numbers for Trees of Small Maximum Degree.

    Google Scholar 

  80. D. R. Heath-Brown, Integer sets containing no arithmetic progressions, J. London Math. Soc. 35 (1987), 385–394.

    Article  MATH  MathSciNet  Google Scholar 

  81. Y. Kohayakawa, The Regularity Lemma of Szemerédi for sparse graphs, manuscript, August 1993.

    Google Scholar 

  82. Y. Kohayakawa, Szemerédi’s regularity lemma for sparse graphs, Foundations of computational mathematics (Rio de Janeiro) (1997), 216–230, Springer, Berlin.

    Google Scholar 

  83. Y. Kohayakawa, B. Kreuter, Threshold functions for asymmetric Ramsey properties involving cycles, Random Structures Algorithms 11 (1997), 245–276.

    Article  MATH  MathSciNet  Google Scholar 

  84. Y. Kohayakawa, T. Luczak, V. Rödl, Arithmetic progressions of length three in subsets of a random set, Acta Arithmetica, 75 (1996), 133–163.

    MATH  MathSciNet  Google Scholar 

  85. Y. Kohayakawa, T. Luczak, V. Rödl, Arithmetic progressions of length three in subsets of a random set, Acta Arith. 75 (1996), 133–163.

    MATH  MathSciNet  Google Scholar 

  86. Y. Kohayakawa, T. Luczak, V. Rödl, On K 4-free subgraphs of random graphs, Combinatorica 17 (1997), 173–213.

    Article  MATH  MathSciNet  Google Scholar 

  87. J. Komlós, The blow-up lemma, Recent trends in combinatorics (Mátraháza, 1995), Combin. Probab. Comput. 8 (1999), 161–176.

    Article  MATH  MathSciNet  Google Scholar 

  88. J. Komlós, Tiling Turán Theorems, Combinatorica 20 (2000), 203–218.

    Article  MATH  MathSciNet  Google Scholar 

  89. J. Komlós, G. N. Sárközy, E. Szemerédi, Proof of a packing conjecture of Bollobás, AMS Conference on Discrete Mathematics, DeKalb, Illinois (1993), Combinatorics, Probability and Computing 4 (1995), 241–255.

    Google Scholar 

  90. J. Komlós, G. N. Sárközy, E. Szemerédi, On the square of a Hamiltonian cycle in dense graphs, Proceedings of the Seventh International Conference on Random Structures and Algorithms (Atlanta, GA, 1995), Random Structures and Algorithms 9 (1996), 193–211.

    Article  Google Scholar 

  91. J. Komlós, G. N. Sárközy, E. Szemerédi, Blow-up Lemma, Combinatorica 17 (1997), 109–123.

    Article  MATH  MathSciNet  Google Scholar 

  92. J. Komlós, G. N. Sárközy, E. Szemerédi, An algorithmic version of the blow-up lemma, Random Structures Algorithms 12 (1998), 297–312.

    Article  MATH  MathSciNet  Google Scholar 

  93. J. Komlós, G. N. Sárközy, E. Szemerédi, On the Pósa-Seymour conjecture, J. Graph Theory 29 (1998), 167–176.

    Article  MATH  MathSciNet  Google Scholar 

  94. János Komlós, Gábor Sárközy, Endre Szemerédi, Proof of the Seymour conjecture for large graphs, Ann. Comb. 2 (1998), 43–60.

    Article  MATH  MathSciNet  Google Scholar 

  95. J. Komlós, G. N. Sárközy, E. Szemerédi, Proof of the Alon-Yuster conjecture, Random Structures and Algorithms.

    Google Scholar 

  96. J. Komlós, M. Simonovits,Szemerédi’s regularity lemma and its applications in graph theory, Bolyai Society Mathematical Studies 2, Combinatorics, Paul Erdős is Eighty (Volume 2) (D. Miklós, V. T. Sós, T. Szőnyi eds.), Keszthely (Hungary) (1993), Budapest (1996), 295–352.

    Google Scholar 

  97. L. Lovász, M. Simonovits, On the number of complete subgraphs of a graph I, Proc. Fifth British Combin. Conf. Aberdeen (1975), 431–442.

    Google Scholar 

  98. L. Lovász, M. Simonovits, On the number of complete subgraphs of a graph II, Studies in Pure Math (dedicated to the memory of P. Turán), Akadémiai Kiadó and Birkhäuser Verlag (1983), 459–495.

    Google Scholar 

  99. T. Luczak, R(C n , C n , C n ) ≤ (4 + o(1))n, J. Combin. Theory B75 (1999), 174–187.

    MathSciNet  Google Scholar 

  100. Luczak, Tomasz; Rödl, Vojtěch; Szemerédi, Endre, Partitioning two-coloured complete graphs into two monochromatic cycles, Combin. Probab. Comput. 7 (1998), 423–436.

    Article  MATH  MathSciNet  Google Scholar 

  101. J. Nešetřil, V. Rödl, Partition theory and its applications, in Surveys in Combinatorics (Proc. Seventh British Combinatorial Conf., Cambridge, 1979), pp. 96–156, (B. Bollobás ed.), London Math. Soc. Lecture Notes Series, Cambridge Univ. Press, Cambridge-New York, 1979.

    Google Scholar 

  102. P. Pudlák, J. Sgall, An upper bound for a communication game, related to time-space tradeoffs, Electronic Colloquium on Computational Complexity, TR 95-010, (1995).

    Google Scholar 

  103. K. F. Roth, On certain sets of integers (II), J. London Math. Soc. 29 (1954), 20–26.

    Article  MATH  MathSciNet  Google Scholar 

  104. K. F. Roth, Irregularities of sequences relative to arithmetic progressions (III), Journal of Number Theory 2 (1970), 125–142.

    Article  MATH  MathSciNet  Google Scholar 

  105. K. F. Roth, Irregularities of sequences relative to arithmetic progressions (IV), Periodica Math. Hung. 2 (1972), 301–326.

    Article  MATH  Google Scholar 

  106. V. Rödl, A generalization of Ramsey Theorem and dimension of graphs, Thesis, 1973, Charles Univ. Prague; see also: A generalization of Ramsey Theorem for graphs, hypergraphs and block systems, Zielona Gora (1976), 211–220.

    Google Scholar 

  107. V. Rödl, On universality of graphs with uniformly distributed edges, Discrete Mathematics 59 (1986), 125–134.

    Article  MATH  MathSciNet  Google Scholar 

  108. V. Rödl, Sparse Regularity, Personal communication.

    Google Scholar 

  109. V. Rödl, A. Ruciński, Random graphs with monochromatic triangles in every edge coloring, Random Structures and Algorithms 5 (1994), 253–270.

    Article  MATH  MathSciNet  Google Scholar 

  110. V. Rödl, A. Ruciński, Threshold functions for Ramsey properties, J. Amer. Math. Soc. 8 (1995), 917–942.

    Article  MATH  MathSciNet  Google Scholar 

  111. I. Z. Ruzsa, E. Szemerédi, Triple systems with no six points carrying three triangles, Combinatorics (Keszthely, 1976), 18 (1978), Vol. II., 939–945, North-Holland, Amsterdam-New York.

    Google Scholar 

  112. G. N. Sárközy, Fast parallel algorithm for finding Hamiltonian cycles and trees in graphs.

    Google Scholar 

  113. P. Seymour, Problem section, Combinatorics: Proceedings of the British Combinatorial Conference 1973 (T. P. McDonough and V. C. Mavron eds.), Cambridge University Press (1974), 201–202.

    Google Scholar 

  114. A. F. Sidorenko, Boundedness of optimal matrices in extremal multigraph and digraph problems, Combinatorica 13 (1993), 109–120.

    Article  MATH  MathSciNet  Google Scholar 

  115. M. Simonovits, A method for solving extremal problems in graph theory, Theory of graphs, Proc. Coll. Tihany (1966) (P. Erdős and G. Katona eds.), Acad. Press, N.Y. (1968), 279–319.

    Google Scholar 

  116. M. Simonovits, Extremal graph problems with symmetrical extremal graphs, additional chromatic conditions, Discrete Mathematics 7 (1974), 349–376.

    MATH  MathSciNet  Google Scholar 

  117. M. Simonovits, Extremal graph theory, Selected Topics in Graph Theory (L. Beineke and R. Wilson eds.) Academic Press, London, New York, San Francisco (1985), 161–200.

    Google Scholar 

  118. M. Simonovits, V. T. Sós, Szemerédi’s partition and quasirandomness, Random Structures and Algorithms 2 (1991), 1–10.

    Article  MATH  MathSciNet  Google Scholar 

  119. M. Simonovits and V. T. Sós, Hereditarily extended properties, quasi-random graphs and not necessarily induced subgraphs, Combinatorica 17 (1997), 577–596.

    Article  MATH  MathSciNet  Google Scholar 

  120. M. Simonovits and V. T. Sós, Hereditarily extended properties, quasi-random graphs and induced subgraphs, to be published.

    Google Scholar 

  121. M. Simonovits and V. T. Sós, Ramsey-Turán theory, Proc. Prague Meeting, Fifth Czech-Slovak International Symposia, Discrete Mathematics, to appear.

    Google Scholar 

  122. V. T. Sós, On extremal problems in graph theory, Proc. Calgary International Conf. on Combinatorial Structures and their Application, Gordon and Breach, N. Y. (1969), 407–410.

    Google Scholar 

  123. V. T. Sós, Interaction of Graph Theory and Number Theory, in Proc. Conf. Paul Erdős and His Mathematics, Budapest, 1999. Springer Verlag, 2001.

    Google Scholar 

  124. E. Szemerédi, On sets of integers containing no four elements in arithmetic progression, Acta Math. Acad. Sci. Hung. 20 (1969), 89–104.

    Article  MATH  Google Scholar 

  125. E. Szemerédi, On graphs containing no complete subgraphs with 4 vertices (in Hungarian), Matematikai Lapok 23 (1972), 111–116.

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  127. E. Szemerédi, Regular partitions of graphs, Colloques Internationaux C.N.R.S. No 260-Problèmes Combinatoires et Théorie des Graphes, Orsay (1976), 399–401.

    Google Scholar 

  128. E. Szemerédi, Integer sets containing no arithmetic progressions, Acta Math. Acad. Sci. Hung. 56 (1990), 155–158.

    MATH  Google Scholar 

  129. A. Thomason, Pseudo-random graphs, in Proc. of Random Graphs, Poznán (1985) (M. Karoński ed.), Annals of Discr. Math. (North-Holland) 33 (1987), 307–331. See also: Dense expanders and bipartite graphs, Discrete Mathematics 75 (1989), 381-386.

    Google Scholar 

  130. Pál Turán, On an extremal problem in graph theory (in Hungarian), Matematikaiés Fizikai Lapok 48 (1941), 436–452.

    MATH  Google Scholar 

  131. B. L. Van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Archief voor Wiskunde 15 (1927), 212–216.

    Google Scholar 

  132. R. Yuster, The number of edge colorings with no monochromatic triangle, J. Graph Theory 21 (1996), 441–452.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Rutgers University, New Brunswick, NJ

    János Komlós, Miklós Simonovits & Endre Szemerédi

  2. Hungarian Academy of Sciences, Hungary

    János Komlós, Miklós Simonovits & Endre Szemerédi

  3. Department of Mathematics and Computer Science, Drexel University, Philadelphia, PA

    Ali Shokoufandeh

Authors
  1. János Komlós
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  2. Ali Shokoufandeh
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  3. Miklós Simonovits
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  4. Endre Szemerédi
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Editors and Affiliations

  1. Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran

    Gholamreza B. Khosrovshahi

  2. Dept. of Mathematics & Computer Science, Drexel University, 19104, Philadelphia, PA, USA

    Ali Shokoufandeh

  3. Digital Fountain, Inc., 39151 Civic Center Drive, 94538, Fremont, CA, USA

    Amin Shokrollahi

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© 2002 Springer-Verlag Berlin Heidelberg

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Komlós, J., Shokoufandeh, A., Simonovits, M., Szemerédi, E. (2002). The Regularity Lemma and Its Applications in Graph Theory. In: Khosrovshahi, G.B., Shokoufandeh, A., Shokrollahi, A. (eds) Theoretical Aspects of Computer Science. TACSci 2000. Lecture Notes in Computer Science, vol 2292. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45878-6_3

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  • DOI: https://doi.org/10.1007/3-540-45878-6_3

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