Skip to main content

The History of Degenerate (Bipartite) Extremal Graph Problems

  • Chapter
Erdős Centennial

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 25))

Abstract

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.

Research supported in part by the Hungarian National Science Foundation OTKA 104343, and by the European Research Council Advanced Investigators Grant 267195 (ZF) and by the Hungarian National Science Foundation OTKA 101536, and by the European Research Council Advanced Investigators Grant 321104. (MS).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Ajtai, J. Komlós, and E. Szemerédi: On a conjecture of Loebl, in Graph theory, Combinatorics, and Algorithms, Vol. 1, 2 (Kalamazoo, MI, 1992), Wiley-Intersci. Publ., pp. 1135–1146. Wiley, New York, 1995.

    Google Scholar 

  2. N. Alon: Eigenvalues and expanders, Combinatorica 6 (1983), 83–96.

    Google Scholar 

  3. N. Alon: Tools from higher algebra, in: „Handbook of Combinatorics“, R. L. Graham, M. Grötschel and L. Lovász, eds, North Holland (1995), Chapter 32, pp. 1749–1783.

    Google Scholar 

  4. N. Alon, S. Hoory, and N. Linial: The Moore bound for irregular graphs, Graphs Combin. 18 (2002), no. 1, 53–57.

    MATH  MathSciNet  Google Scholar 

  5. N. Alon, M. Krivelevich, and B. Sudakov: Turán numbers of bipartite graphs and related Ramsey-type questions, Combin. Probab. Comput. 12 (2003), no. 5–6, 477–494.

    MATH  MathSciNet  Google Scholar 

  6. N. Alon and V. D. Milman: λ1-isoperimetric inequalities for graphs and superconcentrators, J. Combin. Theory Ser. B 38 (1985), 73–88.

    MATH  MathSciNet  Google Scholar 

  7. N. Alon, L. Rónyai, and T. Szabó: Norm-graphs: variations and applications, J. Combin. Theory Ser. B 76 (1999), 280–290.

    MATH  MathSciNet  Google Scholar 

  8. L. Babai and B. Guiduli: Spectral extrema for graphs: the Zarankiewicz problem, Electronic J. Combin. 15 (2009), R123.

    MathSciNet  Google Scholar 

  9. R. Baer: Polarities in finite projective planes, Bull. Amer. Math. Soc. 52 (1946), 77–93.

    MATH  MathSciNet  Google Scholar 

  10. C. Balbuena, P. García-Vázquez, X. Marcote, and J. C. Valenzuela: New results on the Zarankiewicz problem, Discrete Math. 307 (2007), no. 17–18, 2322–2327.

    MATH  MathSciNet  Google Scholar 

  11. C. Balbuena, P. García-Vázquez, X. Marcote, and J. C. Valenzuela: Counterexample to a conjecture of Győri on C 2l -free bipartite graphs, Discrete Math. 307 (2007), no. 6, 748–749.

    MATH  MathSciNet  Google Scholar 

  12. C. Balbuena, P. García-Vázquez, X. Marcote, and J. C. Valenzuela: Extremal K(s; t)-free bipartite graphs, Discrete Math. Theor. Comput. Sci. 10 (2008), no. 3, 35–48.

    MATH  MathSciNet  Google Scholar 

  13. P. N. Balister, B. Bollobás, O. M. Riordan, and R. H. Schelp: Graphs with large maximum degree containing no odd cycles of a given length, J. Combin. Theory B 87 (2003), 366–373.

    MATH  Google Scholar 

  14. P. N. Balister, E. Győri, J. Lehel, and R. H. Schelp: Connected graphs without long paths, Discrete Math 308 (2008), no. 19, 4487–4494.

    MATH  MathSciNet  Google Scholar 

  15. S. Ball and V. Pepe: Asymptotic improvements to the lower bound of certain bipartite Turán numbers, Combin. Probab. Comput. 21 (2012), no. 3, 323–329.

    MATH  MathSciNet  Google Scholar 

  16. J. Beck and J. Spencer: Unit distances, J. Combin. Theory Ser. A 37 (1984), 231–238.

    MATH  MathSciNet  Google Scholar 

  17. F. Behrend: On sets of integers which contain no three terms in arithmetic progression, Proc. Nat. Acad. Sci. US. 32 (1956), 331–332.

    MathSciNet  Google Scholar 

  18. C. T. Benson: Minimal regular graphs of girths eight and twelve, Canad. J. Math. 18 (1966), 1091–1094.

    MATH  MathSciNet  Google Scholar 

  19. D. Bienstock and E. Győri: An extremal problem on sparse 0-1 matrices, SIAM J. Discrete Math. 4 (1991), no. 1, 17–27.

    MATH  MathSciNet  Google Scholar 

  20. P. Blagojević, B. Bukh, and R. Karasev: Turán numbers for Ks,t-free graphs: topological obstructions and algebraic constructions, arXiv:1108.5254v3, 3 Jun 2012.

    Google Scholar 

  21. B. Bollobás: Cycles modulo k, Bull. London Math. Soc. 9 (1977), no. 1, 97–98.

    MATH  MathSciNet  Google Scholar 

  22. B. Bollobás: Extremal Graph Theory, Academic Press, London, 1978.

    MATH  Google Scholar 

  23. B. Bollobás: Random Graphs, Academic Press, London, 1985.

    MATH  Google Scholar 

  24. B. Bollobás: Extremal graph theory, in: R. L. Graham, M. Grötschel, and L. Lovász (Eds.), Handbook of Combinatorics, Elsevier Science, Amsterdam, 1995, pp. 1231–1292.

    Google Scholar 

  25. B. Bollobás and A. Thomason: Proof of a conjecture of Mader, Erdős and Hajnal on topological subgraphs, European J. Combin 19 (1998), 883–887.

    MATH  MathSciNet  Google Scholar 

  26. J. A. Bondy: Basic graph theory: paths and circuits, Handbook of Combinatorics, Vol. I., pp. 3–110, Elsevier, Amsterdam, 1995.

    Google Scholar 

  27. J. A. Bondy: Extremal problems of Paul Erdős on circuits in graphs, Paul Erdős and his mathematics, II (Budapest, 1999), 135–156, Bolyai Soc. Math. Stud., 11, János Bolyai Math. Soc., Budapest, 2002.

    Google Scholar 

  28. J. A. Bondy and M. Simonovits: Cycles or even length in graphs, J. Combin. Theory Ser. B 16 (1974), 97–105.

    MATH  MathSciNet  Google Scholar 

  29. J. A. Bondy and A. Vince: Cycles in a graph whose lengths differ by one or two, J. Graph Theory 27 (1998), 11–15.

    MATH  MathSciNet  Google Scholar 

  30. S. Brandt and E. Dobson: The Erdős-Sós conjecture for graphs of girth 5, Selected papers in honour of Paul Erdős on the occasion of his 80th birthday (Keszthely, 1993), Discrete Math. 150 (1996), no. 1–3. 411–414.

    MATH  MathSciNet  Google Scholar 

  31. P. Brass: Erdős distance problems in normed spaces, Comput. Geom. 6 (1996), no. 4, 195–214.

    MATH  MathSciNet  Google Scholar 

  32. W. G. Brown: On graphs that do not contain a Thomsen graph, Canad. Math. Bull. 9 (1966), 281–285.

    MATH  MathSciNet  Google Scholar 

  33. W. G. Brown: On the non-existence of a type of regular graphs of girth 5, Canad. J. Math. 19 (1967), 644–648.

    MATH  MathSciNet  Google Scholar 

  34. W. G. Brown and J. W. Moon: Sur les ensembles de sommets indépendants dans les graphes chromatiques minimaux, (French), Canad. J. Math. 21 (1969), 274–278.

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  36. W. G. Brown, P. Erdős and V. T. Sós: Some extremal problems on r-graphs, New Directions in the Theory of Graphs (ed. F. Harary), Academic Press, New York, 1973, pp. 53–63.

    Google Scholar 

  37. W. G. Brown and M. Simonovits: Digraph extremal problems, hypergraph extremal problems, and the densities of graph structures, Discrete Math. 48 (1984), no. 2–3, 147–162.

    MATH  MathSciNet  Google Scholar 

  38. W. G. Brown, and M. Simonovits: Extremal multigraph and digraph problems, Paul Erdős and his mathematics, II (Budapest, 1999), pp. 157–203, Bolyai Soc. Math. Stud., 11, János Bolyai Math. Soc., Budapest, 2002.

    Google Scholar 

  39. L. Caccetta and K. Vijayan: Long cycles in subgraphs with prescribed minimum degree, Discrete Math. 97 (1991), no. 1–3, 69–81.

    MATH  MathSciNet  Google Scholar 

  40. D. de Caen and L. A. Székely: The maximum size of 4-and 6-cycle free bipartite graphs on m, n vertices, Sets, Graphs and Numbers (Budapest, 1991), Colloquium Mathematical Society János Bolyai, vol. 60, North-Holland, Amsterdam, 1992, pp. 135–142.

    Google Scholar 

  41. R. Canham: A theorem on arrangements of lines in the plane, Israel J. Math. 7 (1969), 393–397.

    MATH  MathSciNet  Google Scholar 

  42. F. Chung: Subgraphs of a hypercube containing no small even cycles, J. Graph Theory 16 (1992), 273–286.

    MATH  MathSciNet  Google Scholar 

  43. F. R. K. Chung and R. L. Graham: Erdős on Graphs: His Legacy of Unsolved Problems, A. K. Peters Ltd., Wellesley, MA, 1998.

    Google Scholar 

  44. C. R. J. Clapham, A. Flockart, and J. Sheehan: Graphs without four-cycles, J. Graph Theory 13 (1989), 29–47.

    MATH  MathSciNet  Google Scholar 

  45. K. Clarkson, H. Edelsbrunner, L. J. Guibas, M. Sharir, and E. Welzl: Combinatorial complexity bounds for arrangements of curves and spheres, Discrete Comput. Geom. 5 (1990), no. 2, 99–160.

    MATH  MathSciNet  Google Scholar 

  46. M. Conder: Hexagon-free subgraphs of hypercubes, J. Graph Theory 17 (1993), 477–479.

    MATH  MathSciNet  Google Scholar 

  47. D. Conlon: An extremal theorem in the hypercube, Electron. J. Combin. 17 (2010), Research Paper 111.

    Google Scholar 

  48. D. Conlon, J. Fox, and B. Sudakov: An approximate version of Sidorenko’s conjecture, Geom. Funct. Anal. 20 (2010), no. 6, 1354–1366.

    MATH  MathSciNet  Google Scholar 

  49. O. Cooley: Proof of the Loebl-Komlós-Sós conjecture for large, dense graphs, Discrete Math. 309 (2009), no. 21, 6190–6228.

    MATH  MathSciNet  Google Scholar 

  50. K. Čulík: Teilweise Lösung eines verallgemeinerten Problems von K. Zarankiewicz, Ann. Polon. Math. 3 (1956), 165–168.

    MATH  MathSciNet  Google Scholar 

  51. D. M. Cvetkovič, M. Doob, and H. Sachs: Spectra of Graphs, Academic Press Inc., New York, 1980.

    Google Scholar 

  52. G. Damásdi, T. Héger, and T. Szőnyi: The Zarankiewicz problem, cages, and geometries, manuscript 2013.

    Google Scholar 

  53. H. Edelsbrunner and P. Hajnal: A lower bound on the number of unit distances between the vertices of a convex polygon, J. Combin. Theory Ser. A 56 (1991), no. 2, 312–316.

    MATH  MathSciNet  Google Scholar 

  54. P. Erdős: On sequences of integers no one of which divides the product of two others, and some related problems, Mitt. Forschungsinst. Math. u. Mech. Tomsk 2 (1938), 74–82.

    Google Scholar 

  55. P. Erdős: Graph theory and probability I, Canad. J. Math. 11 (1959), 34–38.

    MathSciNet  Google Scholar 

  56. P. Erdős: Graph theory and probability II, Canad. J. Math. 13 (1961), 346–352.

    MathSciNet  Google Scholar 

  57. P. Erdős: Extremal problems in graph theory, Proc. Sympos. Smolenice, 1963, pp. 29–36, Publ. House Czechoslovak Acad. Sci., Prague, 1964.

    Google Scholar 

  58. P. Erdős: Some applications of probability to graph theory and combinatorial problems, Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963), pp. 133–136, Publ. House Czech. Acad. Sci., Prague, 1964.

    Google Scholar 

  59. P. Erdős: On some extremal problems in graph theory, Israel J. Math. 3 (1965), 113–116.

    MathSciNet  Google Scholar 

  60. P. Erdős: Some recent results on extremal problems in graph theory, Theory of Graphs (ed P. Rosenstiehl), (Internat. Sympos., Rome, 1966), Gordon and Breach, New York, and Dunod, Paris, 1967, pp. 117–123.

    Google Scholar 

  61. P. Erdős: On some new inequalities concerning extremal properties of graphs, Theory of Graphs (P. Erdős and G. Katona, Eds.), Academic Press, Nev. York, 1968, pp. 77–81.

    Google Scholar 

  62. P. Erdős: The Art of Counting (ed. J. Spencer), The MIT Press, Cambridge, Mass., 1973.

    Google Scholar 

  63. P. Erdős: Problems and results on finite and infinite combinatorial analysis, in Infinite and Finite Sets (Proc. Conf., Keszthely, Hungary, 1973), pp. 403–424, Proc. Colloq. Math. Soc. J. Bolyai 10, Bolyai-North-Holland, 1975.

    Google Scholar 

  64. P. Erdős: Some recent progress on extremal problems in graph theory, Congr. Numerantium 14 (1975), 3–14.

    Google Scholar 

  65. P. Erdős: Problems and results in combinatorial analysis, Colloquio Internazionale sulle Teorie Combinatorie (Rome, 1973), Tomo II, Atti dei Convegni Lincei, No. 17, pp. 3–17, Accad. Naz. Lincei, Rome, 1976.

    Google Scholar 

  66. P. Erdős: Problems and results in graph theory and combinatorial analysis, Graph theory and related topics (Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1977), pp. 153–163, Academic Press, New York-London.

    Google Scholar 

  67. P. Erdős: On the combinatorial problems which I would most like to see solved, Combinatorica 1 (1981), no. 1, 25–42.

    MathSciNet  Google Scholar 

  68. P. Erdős: On some problems in graph theory, combinatorial analysis and combinatorial number theory, Graph Theory and Combinatorics (Cambridge, 1983), pp. 1–17, Academic Press, London, 1984.

    Google Scholar 

  69. P. Erdős: Two problems in extremal graph theory. Graphs Combin. 2 (1986), no. 1, 189–190.

    MathSciNet  Google Scholar 

  70. P. Erdős: On some of my favourite theorems, Combinatorics, Paul Erdős is eighty, Vol. 2 (Keszthely, 1993), 97–132, Bolyai Soc. Math. Stud., 2, János Bolyai Math. Soc., Budapest, 1996.

    Google Scholar 

  71. P. Erdős, R. J. Faudree, J. Pach, and J. Spencer: How to make a graph bipartite, J. Combin. Theory Ser. B 45 (1988), no. 1, 86–98.

    MathSciNet  Google Scholar 

  72. P. Erdős, R. J. Faudree, R. H. Schelp, and M. Simonovits: An extremal result for paths, Graph theory and its applications: East and West (Jinan, 1986), 155–162 Ann. New York Acad. Sci., 576, New York Acad. Sci., New York, 1989.

    Google Scholar 

  73. P. Erdős, Z. Füredi, M. Loebl, and V. T. Sós: Discrepancy of trees, Studia Sci. Math. Hungar. 30 (1995), no. 1–2, 47–57.

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  75. P. Erdős, E. Győri, and M. Simonovits: How many edges should be deleted to make a triangle-free graph bipartite? Sets, graphs and numbers (Budapest, 1991), pp. 239–263, Colloq. Math. Soc. János Bolyai, 60, North-Holland, Amsterdam, 1992.

    Google Scholar 

  76. P. Erdős, G. Harcos, and J. Pach: Popular distances in 3-space, Discrete Math. 200 (1999), no. 1–3, 95–99.

    MathSciNet  Google Scholar 

  77. P. Erdős and L. Moser: Problem 11, Canad. Math. Bull. 2 (1959), 43.

    Google Scholar 

  78. P. Erdős and A. Rényi: On the evolution of random graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 (1960), 17–61.

    Google Scholar 

  79. P. Erdős and A. Rényi: On a problem in the theory of graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl. 7 (1962), 623–641.

    Google Scholar 

  80. P. Erdős, A. Rényi, and Vera T. Sós: On a problem of graph theory, Stud Sci. Math. Hung. 1 (1966), 215–235.

    Google Scholar 

  81. P. Erdős and H. Sachs: Reguläre Graphen gegebener Taillenweite mit minimaler Knotenzahl (in German), Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 12 (1963), 251–257.

    MathSciNet  Google Scholar 

  82. P. Erdős, A. Sárközy, and V. T. Sós: On product representation of powers, I, European J. Combin. 16 (1995), 567–588.

    Google Scholar 

  83. P. Erdős and M. Simonovits: A limit theorem in graph theory, Studia Sci. Math. Hungar. 1 (1966), 51–57.

    MathSciNet  Google Scholar 

  84. P. Erdős and M. Simonovits: Some extremal problems in graph theory, Combinatorial Theory and Its Applications, I. (Proc. Colloq. Balatonfüred, 1969), North Holland, Amsterdam, 1970, pp. 377–390.

    Google Scholar 

  85. P. Erdős and M. Simonovits: An extremal graph problem, Acta Math. Acad. Sci. Hungar. 22 (1971/72), 275–282.

    MathSciNet  Google Scholar 

  86. P. Erdős and M. Simonovits: Cube-supersaturated graphs and related problems, Progress in Graph Theory (Waterloo, Ont., 1982), pp. 203–218, Academic Press, Toronto, Ont., 1984.

    Google Scholar 

  87. P. Erdős and M. Simonovits: Compactness results in extremal graph theory, Combinatorica 2 (1982), no. 3, 275–288.

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  89. P. Erdős and A. M. Stone: On the structure of linear graphs, Bull. Amer. Math. Soc 52 (1946), 1087–1091.

    MathSciNet  Google Scholar 

  90. G. Fan: Distribution of cycle lengths in graphs, J. Combin. Theory Ser. B 84 (2002), 187–202.

    MATH  MathSciNet  Google Scholar 

  91. G. Fan, Xuezheng Lv, and Pei Wang: Cycles in 2-connected graphs, J. Combin. Theory Ser. B 92 (2004), no. 2, 379–394.

    MATH  MathSciNet  Google Scholar 

  92. R. J. Faudree and R. H. Schelp: Path Ramsey numbers in multicolorings, J. Combin. Theory Ser. B 19 (1975), no. 2, 150–160.

    MATH  MathSciNet  Google Scholar 

  93. R. J. Faudree and M. Simonovits: On a class of degenerate extremal graph problems, Combinatorica 3 (1983), 83–93.

    MATH  MathSciNet  Google Scholar 

  94. J. Fox and B. Sudakov: Dependent random choice, Random Structures Algorithms 38 (2011), no. 1–2, 68–99.

    MATH  MathSciNet  Google Scholar 

  95. Z. Füredi: Graphs without quadrilaterals, J. Combin. Theory Ser. B 34 (1983), 187–190.

    MATH  MathSciNet  Google Scholar 

  96. Z. Füredi: Quadrilateral-free graphs with maximum number of edges, preprint 1988, http://www.math.uiuc.edu/~z-furedi/PUBS/furedi C4from1988.pdf

    Google Scholar 

  97. Z. Füredi: Graphs of diameter 3 with the minimum number of edges, Graphs Combin. 6 (1990), no. 4, 333–337.

    MATH  MathSciNet  Google Scholar 

  98. Z. Füredi: The maximum number of unit distances in a convex n-gon, J. Combin. Theory Ser. A 55 (1990), no. 2, 316–320.

    MATH  MathSciNet  Google Scholar 

  99. Z. Füredi: On a Turán type problem of Erdős, Combinatorica 11 (1991), 75–79.

    MATH  MathSciNet  Google Scholar 

  100. Z. Füredi: Turán type problems, in Surveys in Combinatorics, London Math. Soc. Lecture Note Ser. 166, Cambridge University Press, Cambridge, UK, 1991, pp. 253–300.

    Google Scholar 

  101. Z. Füredi: The maximum number of edges in a minimal graph of diameter 2, J. Graph Theory 16 (1992), no. 1, 81–98.

    MATH  MathSciNet  Google Scholar 

  102. Z. Füredi: Extremal hypergraphs and combinatorial geometry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), pp. 1343–1352, Birkhäuser, Basel, 1995.

    Google Scholar 

  103. Z. Füredi: On the number of edges of quadrilateral-free graphs, J. Combin. Theory Ser. B 68 (1996), 1–6.

    MATH  MathSciNet  Google Scholar 

  104. Z. Füredi: An upper bound on Zarankiewicz problem, Combin. Probab. Comput. 5 (1996), no. 1, 29–33.

    MATH  MathSciNet  Google Scholar 

  105. Z. Füredi: New asymptotics for bipartite Turán numbers, J. Combin. Theory Ser. A 75 (1996), no. 1, 141–144.

    MATH  MathSciNet  Google Scholar 

  106. Z. Füredi and Peter Hajnal: Davenport-Schinzel theory of matrices, Discrete Math. 103 (1992), 231–251.

    Google Scholar 

  107. Z. Füredi, A. Naor, and J. Verstraëte: On the Turán number for the hexagon, Adv. Math. 203 (2006), no. 2, 476–496.

    MATH  MathSciNet  Google Scholar 

  108. Z. Füredi and L. Özkahya: On even-cycle-free subgraphs of the hypercube, J. Combin. Theory Ser. A 118 (2011), 1816–1819.

    MATH  MathSciNet  Google Scholar 

  109. Z. Füredi, O. Pikhurko, and M. Simonovits: The Turán density of the hypergraph {abc; ade; bde; cde}, Electronic J. Combin. 10 (2003), R18.

    Google Scholar 

  110. Z. Füredi and M. Simonovits: Triple systems not containing a Fano configuration, Combin. Probab. Comput. 14 (2005), no. 4, 467–484.

    MATH  MathSciNet  Google Scholar 

  111. Z. Füredi and D. West: Ramsey theory and bandwidth of graphs, Graphs and Combin. 17 (2001), 463–471.

    MATH  MathSciNet  Google Scholar 

  112. D. K. Garnick, Y. H. H. Kwong, and F. Lazebnik: Extremal graphs without threecycles or four-cycles, J. Graph Theory 17 (1993), no. 5, 633–645.

    MATH  MathSciNet  Google Scholar 

  113. D. K. Garnick, and N. A. Nieuwejaar, Non-isomorphic extremal graphs without three-cycles and four-cycles, J. Combin. Math. Combin. Comput. 12 (1992), 33–56.

    MATH  MathSciNet  Google Scholar 

  114. A. Grzesik: On the maximum number of five-cycles in a triangle-free graph, J. Combin. Theory Ser. B 102 (2012), no. 5, 1061–1066.

    MATH  MathSciNet  Google Scholar 

  115. J. R. Griggs and Chih-Chang Ho: On the half-half case of the Zarankiewicz problem, Discrete Math. 249 (2002), no. 1–3, 95–104.

    MATH  MathSciNet  Google Scholar 

  116. J. Griggs, J. Ouyang: (0; 1)-matrices with no half-half submatrix of ones, European J. Combin. 18 (1997), 751–761.

    MATH  MathSciNet  Google Scholar 

  117. J. R. Griggs, M. Simonovits, and George Rubin Thomas: Extremal graphs with bounded densities of small subgraphs, J. Graph Theory 29 (1998), no. 3, 185–207.

    MATH  MathSciNet  Google Scholar 

  118. R. K. Guy: A problem of Zarankiewicz, in: Theory of Graphs (Proc. Colloq., Tihany, 1966), pp. 119–150. Academic Press, New York 1968.

    Google Scholar 

  119. R. K. Guy and S. Znám: A problem of Zarankiewicz, Recent Progress in Combinatorics (Proc. Third Waterloo Conf. on Combinatorics, 1968), pp. 237–243. Academic Press, New York 1969.

    Google Scholar 

  120. A. Gyárfás: Graphs with k odd cycle lengths, Discrete Math. 103 (1992), 41–48.

    MATH  MathSciNet  Google Scholar 

  121. A. Gyárfás, J. Komlós, and E. Szemerédi: On the distribution of cycle lengths in graphs, J. Graph Theory 8 (1984), 441–462.

    MATH  MathSciNet  Google Scholar 

  122. A. Gyárfás, C. C. Rousseau, and R. H. Schelp: An extremal problem for paths in bipartite graphs, J. Graph Theory 8 (1984), 83–95.

    MATH  MathSciNet  Google Scholar 

  123. E. Győri: On the number of C 5’s in a triangle-free graph, Combinatorica 9 (1989), 101–102.

    MathSciNet  Google Scholar 

  124. E. Győri: C 6-free bipartite graphs and product representation of squares, Graphs Combin. (Marseille, 1995), Discrete Math. 165/166 (1997), 371–375.

    Google Scholar 

  125. E. Győri: Triangle-free hypergraphs, Combin. Prob. Comput. 15 (2006), 185–191.

    Google Scholar 

  126. E. Győri, B. Rothschild, and A. Ruciński: Every graph is contained in a sparsest possible balanced graph, Math. Proc. Cambridge Philos. Soc. 98 (1985), no. 3, 397–401.

    MathSciNet  Google Scholar 

  127. R. Häggkvist and A. D. Scott: Arithmetic progressions of cycles, Tech. Rep. Mat. Inst. Umeä Univ. 16, (1998).

    Google Scholar 

  128. S. Hartman, J. Mycielski, C. Ryll-Nardzevski: Systèmes spéciaux de points à coordonn ées entiéres, Colloq. Math. 3 (1954), 84–85, (Bericht Über di Tagung der Poln Math Gesellschaft, Wroclaw, am 20. September 1951.)

    Google Scholar 

  129. H. Hatami: Graph norms and Sidorenko’s conjecture, Israel J. Math. 175 (2010), 125–150.

    MATH  MathSciNet  Google Scholar 

  130. H. Hatami, J. Hladký, D. Král, S. Norine, and A. Razborov: On the number of pentagons in triangle-free graphs, J. Combin. Theory Ser. A 120 (2013), no. 3, 722–732.

    MATH  MathSciNet  Google Scholar 

  131. H. Hatami and S. Norine: Undecidability of linear inequalities in graph homomorphism densities, J. Amer. Math. Soc. 24 (2011), no. 2, 547–565.

    MATH  MathSciNet  Google Scholar 

  132. J. Hladký, J. Komlós, M. Simonovits, M. Stein, and E. Szemerédi: An approximate version of the Loebl-Komlós-Sós Conjecture for sparse graphs, submitted, on arXiv:1211.3050.v1, 2012, Nov 13.

    Google Scholar 

  133. J. Hladký and D. Piguet: Loebl-Komlós-Sós Conjecture: dense case, Manuscript (arXiv:0805:4834).

    Google Scholar 

  134. M. N. Huxley and H. Iwaniec: Bombieri’s theorem in short intervals, Mathematika 22 (1975), 188–194.

    MATH  MathSciNet  Google Scholar 

  135. C. Hyltén-Cavallius: On a combinatorial problem, Colloq. Math. 6 (1958), 59–65.

    MATH  MathSciNet  Google Scholar 

  136. W. Imrich: Explicit construction of graphs without small cycles, Combinatorica 4 (1984), 53–59.

    MATH  MathSciNet  Google Scholar 

  137. C. Jagger, P. Šťovíček, and A. Thomason: Multiplicities of subgraphs, Combinatorica 16 (1996), no. 1, 123–141.

    MATH  MathSciNet  Google Scholar 

  138. S. Janson, T. Luczak, and A. Ruciński: Random Graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience, New York, 2000. xii+333 pp.

    Google Scholar 

  139. T. Jiang: Compact topological minors in graphs, J. Graph Theory 67 (2011), 139–152.

    MATH  MathSciNet  Google Scholar 

  140. T. Jiang and R. Seiver: Turán numbers of subdivided graphs, SIAM J. Discrete Math. 26 (2012), no. 3, 1238–1255.

    MATH  MathSciNet  Google Scholar 

  141. S. Józsa and E. Szemerédi: The number of unit distance on the plane, Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. II, pp. 939–950. Colloq. Math. Soc. János Bolyai, Vol. 10, North-Holland, Amsterdam, 1975.

    Google Scholar 

  142. Gy. Katona, T. Nemetz, and M. Simonovits: On a problem of Turán in the theory of graphs, Mat. Lapok 15 (1964), 228–238.

    MATH  MathSciNet  Google Scholar 

  143. P. Keevash: Hypergraph Turan problems, Surveys in Combinatorics, Cambridge University Press, 2011, 83–140.

    Google Scholar 

  144. P. Keevash and B. Sudakov: The Turan number of the Fano plane, Combinatorica 25 (2005), 561–574.

    MATH  MathSciNet  Google Scholar 

  145. M. Klazar: The Füredi-Hajnal conjecture implies the Stanley-Wilf conjecture, in: D. Krob, A. A. Mikhalev, A. V. Mikhalev (Eds.), Formal Power Series and Algebraic Combinatorics, Springer, Berlin, 2000, pp. 250–255.

    Google Scholar 

  146. J. Kollár, L. Rónyai, and T. Szabó: Norm graphs and bipartite Turán numbers, Combinatorica 16 (1996), 399–406.

    MATH  MathSciNet  Google Scholar 

  147. J. Komlós and E. Szemerédi: Topological cliques in graphs, Combin. Probab. Comput. 3 (1994), no. 2, 247–256.

    MATH  MathSciNet  Google Scholar 

  148. J. Komlós and E. Szemerédi: Topological cliques in graphs II, Combin. Probab. Comput. 5 (1996), 79–90.

    MATH  MathSciNet  Google Scholar 

  149. G. N. Kopylov: Maximal paths and cycles in a graph, Dokl. Akad. Nauk SSSR 234 (1977), no. 1, 19–21. (English translation: Soviet Math. Dokl. 18 (1977), no. 3, 593–596.)

    MathSciNet  Google Scholar 

  150. A. Kostochka and L. Pyber: Small topological complete subgraphs of “dense” graphs, Combinatorica 8 (1988), 83–86.

    MATH  MathSciNet  Google Scholar 

  151. T. Kővári, V. T. Sós, and P. Turán: On a problem of K. Zarankiewicz, Colloq. Math. 3 (1954), 50–57.

    Google Scholar 

  152. D. Kühn and D. Osthus: Four-cycles in graphs without a given even cycle, J. Graph Theory 48 (2005), 147–156.

    MATH  MathSciNet  Google Scholar 

  153. T. Lam and J. Verstraëte: A note on graphs without short even cycles, Electron. J. Combin. 12 (2005), Note 5, 6 pp.

    Google Scholar 

  154. F. Lazebnik and D. Mubayi: New lower bounds for Ramsey numbers of graphs and hypergraphs, Adv. in Appl. Math. 28 (2002), no. 3–4, 544–559.

    MATH  MathSciNet  Google Scholar 

  155. F. Lazebnik and V. A. Ustimenko, New examples of graphs without small cycles and of large size, European J. Combin. 14 (1993), no. 5, 445–460.

    MATH  MathSciNet  Google Scholar 

  156. F. Lazebnik, V. A. Ustimenko, and A. J. Woldar: Properties of certain families of 2k-cycle-free graphs, J. Combin. Theory Ser. B 60 (1994), no. 2, 293–298.

    MATH  MathSciNet  Google Scholar 

  157. F. Lazebnik, V. A. Ustimenko, and A. J. Woldar: A new series of dense graphs of high girth, Bull. Amer. Math. Soc. 32 (1995), no. 1, 73–79.

    MATH  MathSciNet  Google Scholar 

  158. F. Lazebnik, V. A. Ustimenko, and A. J. Woldar, Polarities and 2k-cycle-free graphs, Discrete Math. 197/198 (1999), 503–513.

    MathSciNet  Google Scholar 

  159. F. Lazebnik and A. J.Woldar: General properties of some families of graphs defined by systems of equations, J. Graph Theory 38 (2001), no. 2, 65–86.

    MATH  MathSciNet  Google Scholar 

  160. L. Lovász: Independent sets in critical chromatic graphs, Studia Sci. Math. Hungar. 8 (1973), 165–168.

    MathSciNet  Google Scholar 

  161. L. Lovász: Combinatorial Problems and Exercises, 2nd Ed., North-Holland, Amsterdam, 1993.

    MATH  Google Scholar 

  162. L. Lovász and M. Simonovits: On the number of complete subgraphs of a graph II, Studies in Pure Mathematics, pp. 459–495, (dedicated to the memory of P. Turán), Akadémiai Kiadó and Birkhäuser Verlag 1982.

    Google Scholar 

  163. A. A. Razborov: Flag algebras, J. Symbolic Logic 72 (2007), no. 4, 1239–1282.

    MATH  MathSciNet  Google Scholar 

  164. A. Lubotzky, R. Phillips, and P. Sarnak: Ramanujan graphs, Combinatorica 8 (1988), no. 3, 261–277.

    MATH  MathSciNet  Google Scholar 

  165. A. McLennan: The Erdős-Sós conjecture for trees of diameter four, J. Graph Theory 49 (2005), no. 4, 291–301.

    MATH  MathSciNet  Google Scholar 

  166. W. Mader: Homomorphieeigenschaften und mittlere Kantendichte von Graphen, Math. Ann. 174 (1967), 265–268.

    MATH  MathSciNet  Google Scholar 

  167. W. Mader: Topological subgraphs in graphs of large girth, Combinatorica 18 (1998), no. 3, 405–412.

    MATH  MathSciNet  Google Scholar 

  168. W. Mader: Topological minors in graphs of minimum degree n, Contemporary trends in discrete mathematics (Štiřín Castle, 1997), 199–211, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 49, Amer. Math. Soc., Providence, RI, 1999.

    Google Scholar 

  169. W. Mader: Graphs with 3n — 6 edges not containing a subdivision of K 5, Combinatorica 25 (2005), no. 4, 425–438.

    MATH  MathSciNet  Google Scholar 

  170. A. Marcus and G. Tardos: Excluded permutation matrices and the Stanley-Wilf conjecture, J. Combin. Theory Ser. A 107 (2004), no. 1, 153–160.

    MATH  MathSciNet  Google Scholar 

  171. G. A. Margulis: Explicit construction of graphs without short cycles and low density codes, Combinatorica 2 (1982), 71–78.

    MATH  MathSciNet  Google Scholar 

  172. G. A. Margulis: Arithmetic groups and graphs without short cycles, in: 6th Int. Symp. on Information Theory, Tashkent, Abstracts 1, 1984, pp. 123–125 (in Russian).

    Google Scholar 

  173. G. A. Margulis: Explicit group-theoretical construction of combinatorial schemes and their application to the design of expanders and concentrators, J. Problems of Inform. Trans. 24 (1988), 39–46; translation from Problemy Peredachi Informatsii 24 (January-March 1988), 51–60.

    MATH  MathSciNet  Google Scholar 

  174. G. Megyesi and E. Szabó: On the tacnodes of configurations of conics in the projective plane, Math. Ann. 305 (1996), no. 4, 693–703.

    MATH  MathSciNet  Google Scholar 

  175. M. Molloy and B. Reed: Graph Colouring and the Probabilistic Method, Algorithms and Combinatorics, 23. Springer-Verlag, Berlin, 2002, xiv+326 pp.

    Google Scholar 

  176. B. Montágh: Unavoidable substructures, PHD Thesis, University of Memphis, May 2005.

    Google Scholar 

  177. M. Mörs: A new result on the problem of Zarankiewicz, J. Combin. Theory Ser. A 31 (1981), no. 2, 126–130.

    MATH  MathSciNet  Google Scholar 

  178. D. Mubayi and Gy. Turán: Finding bipartite subgraphs efficiently, Inform. Process. Lett. 110 (2010), no. 5, 174–177.

    MATH  MathSciNet  Google Scholar 

  179. Z. L. Nagy: A multipartite version of the Turán problem — density conditions and eigenvalues, Electron. J. Combin. 18 (2011), no. 1, Paper 46, 15 pp.

    Google Scholar 

  180. V. Nikiforov: Bounds on graph eigenvalues II, Linear Algebra Appl. 427 (2007), 183–189.

    MATH  MathSciNet  Google Scholar 

  181. V. Nikiforov: A contribution to the Zarankiewicz problem, Linear Algebra Appl. 432 (2010), no. 6, 1405–1411.

    MATH  MathSciNet  Google Scholar 

  182. J. Pach and P. K. Agarwal: Combinatorial Geometry, Wiley-Interscience, New York, 1995. xiv+354 pp.

    MATH  Google Scholar 

  183. D. Piguet and M. J. Stein: Loebl-Komlós-Sós conjecture for trees of diameter 5, Electron. J. Combin., 15 (2008), Research Paper 106, 11 pp.

    Google Scholar 

  184. D. Piguet and M. J. Stein: An approximate version of the Loebl-Komlós-Sós conjecture, J. Combin. Theory Ser. B 102 (2012), no. 1, 102–125.

    MATH  MathSciNet  Google Scholar 

  185. O. Pikhurko: A note on the Turán Function of even cycles, Proc. Amer. Math Soc. 140 (2012), 3687–3992.

    MATH  MathSciNet  Google Scholar 

  186. R. Pinchasi and M. Sharir: On graphs that do not contain the cube and related problems, Combinatorica 25 (2005), no. 5, 615–623.

    MATH  MathSciNet  Google Scholar 

  187. I. Reiman: Über ein Problem von K. Zarankiewicz, Acta Math. Acad. Sci. Hungar. 9 (1958), no. 3–4, 269–273.

    MATH  MathSciNet  Google Scholar 

  188. I. Reiman: An extremal problem in graph theory, (Hungarian). Mat. Lapok 12 (1961), 44–53.

    MATH  MathSciNet  Google Scholar 

  189. A. Rényi: Selected Papers of Alfréd Rényi, Akadémiai Kiadó, 1976 (ed. Paul Turán).

    Google Scholar 

  190. J.-F. Saclè and M. Woźniak: A note on the Erdős-Sós conjecture for graphs without C 4, J. Combin. Theory Ser. B 70 (1997), no. 2, 367–372.

    MATH  MathSciNet  Google Scholar 

  191. G. N. Sárközy: Cycles in bipartite graphs and an application in number theory, J. Graph Theory, 19 (1995), 323–331.

    MATH  MathSciNet  Google Scholar 

  192. A. Scott: Szemerédi’s regularity lemma for matrices and sparse graphs, Combin. Probab. Comput. 20 (2011), no. 3, 455–466.

    MATH  MathSciNet  Google Scholar 

  193. Jian Shen: On two Turán numbers, J. Graph Theory 51 (2006), 244–250.

    MATH  MathSciNet  Google Scholar 

  194. A. F. Sidorenko: Asymptotic solution for a new class of forbidden r-graphs, Combinatorica 9 (1989), no. 2, 207–215.

    MATH  MathSciNet  Google Scholar 

  195. A. Sidorenko: A correlation inequality for bipartite graphs, Graphs Combin. 9 (1993), no. 2, 201–204.

    MATH  MathSciNet  Google Scholar 

  196. A. F. Sidorenko: What do we know and what we do not know about Turán Numbers, Graphs Combin. 11 (1995), no. 2, 179–199.

    MATH  MathSciNet  Google Scholar 

  197. M. Simonovits: A method for solving extremal problems in graph theory, Theory of Graphs, Proc. Colloq. Tihany, (1966), (P. Erdős and G. Katona, Eds.), pp. 279–319, Acad. Press, New York, 1968.

    Google Scholar 

  198. M. Simonovits: On colour-critical graphs, Studia Sci. Math. Hungar. 7 (1972), 67–81.

    MATH  MathSciNet  Google Scholar 

  199. M. Simonovits: Note on a hypergraph extremal problem, Hypergraph Seminar, Columbus Ohio USA, 1972, (C. Berge and D. K. Ray-Chaudhuri, Eds.), Lecture Notes in Mathematics 411, pp. 147–151, Springer Verlag, 1974.

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  201. M. Simonovits: On Paul Turán’s influence on graph theory, J. Graph Theory 1 (1977), no. 2, 102–116.

    MATH  MathSciNet  Google Scholar 

  202. M. Simonovits: Extremal graph problems and graph products, Studies in Pure Mathematics, pp. 669–680, (dedicated to the memory of P. Turán), Akadémiai Kiadó and Birkhäuser Verlag 1982.

    Google Scholar 

  203. M. Simonovits: Extremal graph theory, in: L. W. Beineke, R. J. Wilson (Eds.), Selected Topics in Graph Theory II., pp. 161–200, Academic Press, London, 1983.

    Google Scholar 

  204. M. Simonovits: Extremal graph problems, degenerate extremal problems and supersaturated graphs, Progress in graph Theory, (Bondy and Murty, Eds.), pp. 419–438, Academic Press, 1984.

    Google Scholar 

  205. M. Simonovits: How to solve a Turán type extremal graph problem? (linear decomposition), Contemporary trends in discrete mathematics (Stirin Castle, 1997), pp. 283–305, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 49, Amer. Math. Soc., Providence, RI, 1999.

    Google Scholar 

  206. M. Simonovits: Paul Erdős’ influence on extremal graph theory, The mathematics of Paul Erdős, II., pp. 148–192, Algorithms Combin., 14, Springer, Berlin, 1997.

    Google Scholar 

  207. M. Simonovits and V. T. Sós: Ramsey-Turán theory, Combinatorics, graph theory, algorithms and applications, Discrete Math. 229 (2001), no. 1–3, 293–340.

    MATH  MathSciNet  Google Scholar 

  208. R. R. Singleton: On minimal graphs of maximum even girth, J. Combinatorial Theory 1 (1966), 306–332.

    MATH  MathSciNet  Google Scholar 

  209. V. T. Sós: Remarks on the connection of graph theory, finite geometry and block designs, Colloquio Internazionale sulle Teorie Combinatorie (Roma, 1973), Tomo II, pp. 223–233, Atti dei Convegni Lincei, No. 17, Accad. Naz. Lincei, Rome, 1976.

    Google Scholar 

  210. J. Spencer, E. Szemerédi, and W. T. Trotter: Unit distances in the Euclidean plane, Graph theory and combinatorics (Cambridge, 1983), pp. 293–303, Academic Press, London, 1984.

    Google Scholar 

  211. B. Sudakov and J. Verstraëte: Cycle lengths in sparse graphs, Combinatorica 28 (2008), no. 3, 357–372.

    MATH  MathSciNet  Google Scholar 

  212. E. Szemerédi: Regular partitions of graphs, Problemes Combinatoires et Theorie des Graphes (ed. I.-C. Bermond et al.), pp. 399–401, CNRS, Paris, 1978.

    Google Scholar 

  213. G. Tardos: On 0-1 matrices and small excluded submatrices, J. Combin. Th. Ser. A 111 (2005), 266–288.

    MATH  MathSciNet  Google Scholar 

  214. A. G. Thomason: A disproof of a conjecture of Erdős in Ramsey Theory, J. London Math. Soc. 39 (1989), 246–255.

    MATH  MathSciNet  Google Scholar 

  215. A. Thomason and P. Wagner: Bounding the size of square-free subgraphs of the hypercube, Discrete Math. 309 (2009), 1730–1735.

    MATH  MathSciNet  Google Scholar 

  216. C. M. Timmons: Ordered Turán Problems, Lecture no. 1086-05-1067 on the Joint Mathematics Meetings, San Diego, CA, January 9, 2013.

    Google Scholar 

  217. B. Toft: Two theorems on critical 4-chromatic graphs, Studia Sci. Math. Hungar. 7 (1972), 83–89.

    MATH  MathSciNet  Google Scholar 

  218. P. Turán: On a theorem of Hardy-Ramanujan, Journal of London Math Soc. 9 (1934), 274–276.

    Google Scholar 

  219. P. Turán: On an extremal problem in graph theory, (Hungarian), Mat. Fiz. Lapok 48 (1941), 436–452.

    MathSciNet  Google Scholar 

  220. P. Turán: On the theory of graphs, Colloq. Math. 3 (1954), 19–30.

    MATH  Google Scholar 

  221. P. Turán: A note of welcome, J. Graph Theory 1 (1977), 7–9.

    Google Scholar 

  222. J. Verstraëte: On arithmetic progressions of cycle lengths in graphs, Combin. Probab. Comput. 9 (2000), no. 4, 369–373.

    MATH  MathSciNet  Google Scholar 

  223. R. Wenger: Extremal graphs with no C 4’s, C 6’s, or C 10’s, J. Combin. Theory Ser. B 52 (1991), no. 1, 113–116.

    MATH  MathSciNet  Google Scholar 

  224. R. M. Wilson: An existence theory for pairwise balanced designs, III. Proof of the existence conjectures, J. Combin. Theory Ser. A 18 (1975), 71–79.

    MATH  Google Scholar 

  225. D. R. Woodall: Maximal circuits of graphs I, Acta Math. Acad. Sci. Hungar. 28 (1976), no. 1–2, 77–80.

    MATH  MathSciNet  Google Scholar 

  226. D. R. Woodall: Maximal circuits of graphs II, Studia Sci. Math. Hungar. 10 (1975), no. 1–2, 103–109.

    MathSciNet  Google Scholar 

  227. M. Woźniak: On the Erdős-Sós conjecture, J. Graph Theory, 21 (1996), no. 2, 229–234.

    MATH  MathSciNet  Google Scholar 

  228. Y. Yuansheng and P. Rowlinson: On extremal graphs without four-cycles, Utilitas Math. 41 (1992), 204–210.

    MATH  MathSciNet  Google Scholar 

  229. Y. Yuansheng and P. Rowlinson: On graphs without 6-cycles and related Ramsey numbers, Utilitas Math. 44 (1993), 192–196.

    MATH  MathSciNet  Google Scholar 

  230. K. Zarankiewicz: Problem 101, Colloquium Mathematicum 2 (1951), p. 301.

    Google Scholar 

  231. Yi Zhao: Proof of the (n=2-n=2-n=2) conjecture for large n, Electron. J. Combin. 18 (2011), Paper 27.

    Google Scholar 

  232. Š. Znám: On a combinatorical problem of K. Zarankiewicz, Colloq. Math. 11 (1963), 81–84.

    MATH  MathSciNet  Google Scholar 

  233. Š. Znám: Two improvements of a result concerning a problem of K. Zarankiewicz, Colloq. Math. 13 (1964/1965), 255–258.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2013 János Bolyai Mathematical Society and Springer-Verlag

About this chapter

Cite this chapter

Füredi, Z., Simonovits, M. (2013). The History of Degenerate (Bipartite) Extremal Graph Problems. In: Lovász, L., Ruzsa, I.Z., Sós, V.T. (eds) Erdős Centennial. Bolyai Society Mathematical Studies, vol 25. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39286-3_7

Download citation

Publish with us

Policies and ethics