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

# The Regularity Lemma and Its Applications in Graph Theory

• János Komlós
• Ali Shokoufandeh
• Miklós Simonovits
• Endre Szemerédi
Chapter

## 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.

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### References

1. [1]
M. Ajtai, J. Komlós, M. Simonovits and E. Szemerédi, Solution of the Erdős-Sós Conjecture, in preparation.Google Scholar
2. [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.
3. [3]
N. Alon, E. Fischer, 2-factors in dense graphs, Discrete Math.Google Scholar
4. [4]
N. Alon, R. Yuster, Almost H-factors in dense graphs, Graphs and Combinatorics 8 (1992), 95–102.
5. [5]
N. Alon, R. Yuster, H-factors in dense graphs, J. Combinatorial Theory B66 (1996), 269–282.
6. [6]
L. Babai, M. Simonovits, J. Spencer, Extremal subgraphs of random graphs, Journal of Graph Theory 14 (1990), 599–622.
7. [7]
V. Bergelson, A. Leibman, Polynomial extension of van der Waerden’s and Szemerédi’s theorem.Google Scholar
8. [8]
B. Bollobás, Extremal graph theory, Academic Press, London (1978).
9. [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. [10]
B. Bollobás, P. Erdős, On a Ramsey-Turán type problem, Journal of Combinatorial Theory B21 (1976), 166–168.
11. [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.
12. [12]
B. Bollobás, A. Thomason, The structure of hereditary properties and colourings of random graphs. Combinatorica 20 (2000), 173–202.
13. [13]
W. G. Brown, P. Erdős, M. Simonovits, Extremal problems for directed graphs, Journal of Combinatorial Theory B15 (1973), 77–93.
14. [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. [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.
16. [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. [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.
18. [18]
W. G. Brown, M. Simonovits, Digraph extremal problems, hypergraph extremal problems, and densities of graph structures, Discrete Mathematics 48 (1984), 147–162.
19. [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. [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.
21. [21]
L. Caccetta, R. Häggkvist, On diameter critical graphs, Discrete Mathematics 28 (1979), 223–229.
22. [22]
Fan R. K. Chung, Regularity lemmas for hypergraphs and quasi-randomness, Random Structures and Algorithms 2 (1991), 241–252.
23. [23]
F. R. K. Chung, R. L. Graham, R. M. Wilson, Quasi-random graphs, Combinatorica 9 (1989), 345–362.
24. [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.
25. [25]
V. Chvátal, E. Szemerédi, On the Erdős-Stone theorem, Journal of the London Math. Soc. 23 (1981), 207–214.
26. [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.
27. [27]
K. Corrádi, A. Hajnal, On the maximal number of independent circuits in a graph, Acta Math. Acad. Sci. Hung. 14 (1963), 423–439.
28. [28]
W. Deuber, Generalizations of Ramsey’s theorem, Proc. Colloq. Math. Soc. János Bolyai 10 (1974), 323–332.
29. [29]
G. A. Dirac, Some theorems on abstract graphs, Proc. London Math. Soc. 2 (1952), 68–81.
30. [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.
31. [31]
R. A. Duke, V. Rödl, On graphs with small subgraphs of large chromatic number, Graphs Combin. 1 (1985), 91–96.
32. [32]
R. A. Duke, V. Rödl, The Erdös-Ko-Rado theorem for small families, J. Combin. Theory Ser. A65 (1994), 246–251.
33. [33]
P. Erdős, Some recent results on extremal problems in graph theory, Results, International Symposium, Rome (1966), 118–123.Google Scholar
34. [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. [35]
P. Erdős, On some extremal problems on r-graphs, Discrete Mathematics 1 (1971), 1–6.
36. [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. [37]
P. Erdős, On the combinatorial problems which I would most like to see solved, Combinatorica 1 (1981), 25–42.
38. [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.
39. [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. [40]
P. Erdős, T. Gallai, On maximal paths and circuits of graphs, Acta Math. Acad. Sci. Hung. 10 (1959), 337–356.
41. [41]
P. Erdős, A. Hajnal, On complete topological subgraphs of certain graphs, Annales Univ. Sci. Budapest 7 (1969), 193–199.Google Scholar
42. [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. [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.
44. [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. [45]
P. Erdős, A. Hajnal, V. T. Sós, E. Szemerédi, More results on Ramsey-Turán type problems, Combinatorica 3 (1983), 69–81.
46. [46]
P. Erdős, M. Simonovits, A limit theorem in graph theory, Studia Sci. Math. Hung. 1 (1966), 51–57.Google Scholar
47. [47]
P. Erdős, M. Simonovits, The chromatic properties of geometric graphs, Ars Combinatoria 9 (1980), 229–246.
48. [48]
P. Erdős, M. Simonovits, Supersaturated graphs and hypergraphs, Combinatorica 3 (1983), 181–192.
49. [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. [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. [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. [52]
P. Erdős, A. H. Stone, On the structure of linear graphs, Bull. Amer. Math. Soc. 52 (1946), 1089–1091.Google Scholar
53. [53]
P. Erdős, P. Turán, On some sequences of integers, J. London Math. Soc. 11 (1936), 261–264.
54. [54]
G. Fan, R. Häggkvist, The square of a hamiltonian cycle, SIAM J. Disc. Math.Google Scholar
55. [55]
G. Fan, H. A. Kierstead, The square of paths and cycles, Journal of Combinatorial Theory B63 (1995), 55–64.
56. [56]
G. Fan, H. A. Kierstead, The square of paths and cycles II.Google Scholar
57. [57]
R. J. Faudree, R. J. Gould, M. Jacobson, On a problem of Pósa and Seymour.Google Scholar
58. [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. [59]
P. Frankl, Z. Füredi, Exact solution of some Turán-type problems, Journal of Combinatorial Theory A45 (1987), 226–262.Google Scholar
60. [60]
P. Frankl, J. Pach, An extremal problem on Kr-free graphs, Journal of Graph Theory 12 (1988), 519–523.
61. [61]
P. Frankl, V. Rödl, The Uniformity Lemma for hypergraphs, Graphs and Combinatorics 8 (1992), 309–312.
62. [62]
Alan Frieze, Ravi Kannan, Quick approximation to matrices and applications, Combinatorica 19 (1999), 175–220.
63. [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. [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. [65]
Z. Füredi, The maximum number of edges in a minimal graph of diameter 2, Journal of Graph Theory 16 (1992), 81–98.
66. [66]
H. Fcurstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, Journal d’Analyse Math. 31 (1977), 204–256.
67. [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, 1996Google Scholar
68. [68]
H. Fürstenberg, Y. Katznelson, Idempotents in compact semigroups and Ramsey theory, Israel Journal of Mathematics 68 (1989), 257–270.
69. [69]
H. Fürstenberg, Y. Katznelson, A density version of the Hales-Jewett theorem, Journal d'Analyse Math. 57 (1991), 64–119.Google Scholar
70. [70]
W. T. Gowers, Lower bounds of tower type for Szemerédi’s uniformity lemma, Geom. Funct. Anal. 7 (1997), 322–337.
71. [71]
R. L. Graham, B. L. Rothschild, J. Spencer, Ramsey Theory, Wiley Interscience, Series in Discrete Mathematics (1980).Google Scholar
72. [72]
A. Hajnal, W. Maass, Gy. Turán, On the communication complexity of graph properties, 20th STOC, Chicago (1988), 186–191.Google Scholar
73. [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. [74]
P. E. Haxell, Y. Kohayakawa, The size-Ramsey number of trees, Israel J. Math. 89 (1995), 261–274.
75. [75]
P. E. Haxell, Y. Kohayakawa, On an anti-Ramsey property of Ramanujan graphs, Random Structures and Algorithms 6 (1995), 417–431.
76. [76]
P. E. Haxell, Y. Kohayakawa, T. Luczak, The induced size-Ramsey number of cycles, Combinatorics, Probability and Computing 4 (1995), 217–239.
77. [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.
78. [78]
P. E. Haxell, Y. Kohayakawa, T. Luczak, Turán’s extremal problem in random graphs: forbidding odd cycles, Combinatorica 16 (1996), 107–122.
79. [79]
P. E. Haxell, T. Luczak, P. W. Tingley, Ramsey Numbers for Trees of Small Maximum Degree.Google Scholar
80. [80]
D. R. Heath-Brown, Integer sets containing no arithmetic progressions, J. London Math. Soc. 35 (1987), 385–394.
81. [81]
Y. Kohayakawa, The Regularity Lemma of Szemerédi for sparse graphs, manuscript, August 1993.Google Scholar
82. [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. [83]
Y. Kohayakawa, B. Kreuter, Threshold functions for asymmetric Ramsey properties involving cycles, Random Structures Algorithms 11 (1997), 245–276.
84. [84]
Y. Kohayakawa, T. Luczak, V. Rödl, Arithmetic progressions of length three in subsets of a random set, Acta Arithmetica, 75 (1996), 133–163.
85. [85]
Y. Kohayakawa, T. Luczak, V. Rödl, Arithmetic progressions of length three in subsets of a random set, Acta Arith. 75 (1996), 133–163.
86. [86]
Y. Kohayakawa, T. Luczak, V. Rödl, On K 4-free subgraphs of random graphs, Combinatorica 17 (1997), 173–213.
87. [87]
J. Komlós, The blow-up lemma, Recent trends in combinatorics (Mátraháza, 1995), Combin. Probab. Comput. 8 (1999), 161–176.
88. [88]
J. Komlós, Tiling Turán Theorems, Combinatorica 20 (2000), 203–218.
89. [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. [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.
91. [91]
J. Komlós, G. N. Sárközy, E. Szemerédi, Blow-up Lemma, Combinatorica 17 (1997), 109–123.
92. [92]
J. Komlós, G. N. Sárközy, E. Szemerédi, An algorithmic version of the blow-up lemma, Random Structures Algorithms 12 (1998), 297–312.
93. [93]
J. Komlós, G. N. Sárközy, E. Szemerédi, On the Pósa-Seymour conjecture, J. Graph Theory 29 (1998), 167–176.
94. [94]
János Komlós, Gábor Sárközy, Endre Szemerédi, Proof of the Seymour conjecture for large graphs, Ann. Comb. 2 (1998), 43–60.
95. [95]
J. Komlós, G. N. Sárközy, E. Szemerédi, Proof of the Alon-Yuster conjecture, Random Structures and Algorithms.Google Scholar
96. [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. [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. [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. [99]
T. Luczak, R(C n, C n, C n) ≤ (4 + o(1))n, J. Combin. Theory B75 (1999), 174–187.
100. [100]
Luczak, Tomasz; Rödl, Vojtěch; Szemerédi, Endre, Partitioning two-coloured complete graphs into two monochromatic cycles, Combin. Probab. Comput. 7 (1998), 423–436.
101. [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. [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. [103]
K. F. Roth, On certain sets of integers (II), J. London Math. Soc. 29 (1954), 20–26.
104. [104]
K. F. Roth, Irregularities of sequences relative to arithmetic progressions (III), Journal of Number Theory 2 (1970), 125–142.
105. [105]
K. F. Roth, Irregularities of sequences relative to arithmetic progressions (IV), Periodica Math. Hung. 2 (1972), 301–326.
106. [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. [107]
V. Rödl, On universality of graphs with uniformly distributed edges, Discrete Mathematics 59 (1986), 125–134.
108. [108]
V. Rödl, Sparse Regularity, Personal communication.Google Scholar
109. [109]
V. Rödl, A. Ruciński, Random graphs with monochromatic triangles in every edge coloring, Random Structures and Algorithms 5 (1994), 253–270.
110. [110]
V. Rödl, A. Ruciński, Threshold functions for Ramsey properties, J. Amer. Math. Soc. 8 (1995), 917–942.
111. [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. [112]
G. N. Sárközy, Fast parallel algorithm for finding Hamiltonian cycles and trees in graphs.Google Scholar
113. [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. [114]
A. F. Sidorenko, Boundedness of optimal matrices in extremal multigraph and digraph problems, Combinatorica 13 (1993), 109–120.
115. [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. [116]
M. Simonovits, Extremal graph problems with symmetrical extremal graphs, additional chromatic conditions, Discrete Mathematics 7 (1974), 349–376.
117. [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. [118]
M. Simonovits, V. T. Sós, Szemerédi’s partition and quasirandomness, Random Structures and Algorithms 2 (1991), 1–10.
119. [119]
M. Simonovits and V. T. Sós, Hereditarily extended properties, quasi-random graphs and not necessarily induced subgraphs, Combinatorica 17 (1997), 577–596.
120. [120]
M. Simonovits and V. T. Sós, Hereditarily extended properties, quasi-random graphs and induced subgraphs, to be published.Google Scholar
121. [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. [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. [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. [124]
E. Szemerédi, On sets of integers containing no four elements in arithmetic progression, Acta Math. Acad. Sci. Hung. 20 (1969), 89–104.
125. [125]
E. Szemerédi, On graphs containing no complete subgraphs with 4 vertices (in Hungarian), Matematikai Lapok 23 (1972), 111–116.Google Scholar
126. [126]
E. Szemerédi, On sets of integers containing no k elements in arithmetic progression, Acta Arithmetica 27 (1975), 199–245.
127. [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. [128]
E. Szemerédi, Integer sets containing no arithmetic progressions, Acta Math. Acad. Sci. Hung. 56 (1990), 155–158.
129. [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. [130]
Pál Turán, On an extremal problem in graph theory (in Hungarian), Matematikaiés Fizikai Lapok 48 (1941), 436–452.
131. [131]
B. L. Van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Archief voor Wiskunde 15 (1927), 212–216.Google Scholar
132. [132]
R. Yuster, The number of edge colorings with no monochromatic triangle, J. Graph Theory 21 (1996), 441–452.

## Authors and Affiliations

• János Komlós
• 1
• 2
• Ali Shokoufandeh
• 3
• Miklós Simonovits
• 1
• 2
• Endre Szemerédi
• 1
• 2
1. 1.Rutgers UniversityNew Brunswick