On the Fon-Der-Flaass interpretation of extremal examples for Turán’s (3, 4)-problem

  • Alexander A. Razborov


Fon-Der-Flaass (1988) presented a general construction that converts an arbitrary \(\vec C_4 \)-free oriented graph Γ into a Turán (3, 4)-graph. He observed that all Turán-Brown-Kostochka examples result from his construction, and proved the lower bound \(\tfrac{4} {9} \) (1 − o(1)) on the edge density of any Turán (3, 4)-graph obtainable in this way. In this paper we establish the optimal bound \(\tfrac{3} {7} \) (1 − o(1)) on the edge density of any Turán (3, 4)-graph resulting from the Fon-Der-Flaass construction under any of the following assumptions on the undirected graph G underlying the oriented graph Γ: (i) G is complete multipartite; (ii) the edge density of G is not less than \(\tfrac{2} {3} - \varepsilon \) for some absolute constant ε > 0. We are also able to improve Fon-Der-Flaass’s bound to \(\tfrac{7} {{16}} \) (1 − o(1)) without any extra assumptions on Γ.


STEKLOV Institute Edge Density Oriented Graph Extra Assumption Free Vertex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    D. G. Fon-Der-Flaass, “Method for Construction of (3, 4)-Graphs,” Mat. Zametki 44(4), 546–550 (1998) [Math. Notes 44, 781–783 (1988)].MathSciNetGoogle Scholar
  2. 2.
    W. G. Brown, “On an Open Problem of Paul Turán Concerning 3-Graphs,” in Studies in Pure Mathematics (Birkhäuser, Basel, 1983), pp. 91–93.Google Scholar
  3. 3.
    D. de Caen, “The Current Status of Turán’s Problem on Hypergraphs,” in Extremal Problems for Finite Sets, Visegrád, Hungary, 1991 (János Bolyai Math. Soc., Budapest, 1994), Bolyai Soc. Math. Stud. 3, pp. 187–197.Google Scholar
  4. 4.
    F. Chung and L. Lu, “An Upper Bound for the Turán Number t 3(n, 4),” J. Comb. Theory A 87, 381–389 (1999).MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    D. C. Fisher, “Lower Bounds on the Number of Triangles in a Graph,” J. Graph Theory 13(4), 505–512 (1989).MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    A. W. Goodman, “On Sets of Acquaintances and Strangers at Any Party,” Am. Math. Mon. 66(9), 778–783 (1959).zbMATHCrossRefGoogle Scholar
  7. 7.
    A. V. Kostochka, “A Class of Constructions for Turán’s (3, 4)-Problem,” Combinatorica 2(2), 187–192 (1982).MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    L. Lovász and M. Simonovits, “On the Number of Complete Subgraphs of a Graph. II,” in Studies in Pure Mathematics (Birkhäuser, Basel, 1983), pp. 459–495.Google Scholar
  9. 9.
    L. Lovász and B. Szegedy, “Limits of Dense Graph Sequences,” J. Comb. Theory B 96(6), 933–957 (2006).zbMATHCrossRefGoogle Scholar
  10. 10.
    W. Mantel, “Vraagstuk XXVIII,” Wiskundige Opgaven 10, 60–61 (1907).Google Scholar
  11. 11.
    V. Nikiforov, “The Number of Cliques in Graphs of Given Order and Size,” arXiv: 0710.2305v2.Google Scholar
  12. 12.
    O. Pikhurko, “The Minimum Size of 3-Graphs without a 4-Set Spanning No or Exactly Three Edges,” Manuscript (Carnegie Mellon Univ., Pittsburgh, 2009), Google Scholar
  13. 13.
    A. A. Razborov, “Flag Algebras,” J. Symb. Log. 72(4), 1239–1282 (2007).MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    A. A. Razborov, “On the Minimal Density of Triangles in Graphs,” Comb. Probab. Comput. 17(4), 603–618 (2008).MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    A. A. Razborov, “On 3-Hypergraphs with Forbidden 4-Vertex Configurations,” SIAM J. Discrete Math. 24(3), 946–963 (2010).MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    A. Sidorenko, “What We Know and What We Do Not Know about Turán Numbers,” Graphs Comb. 11, 179–199 (1995).MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    P. Turán, “Egy gráfelméleti szélsöértékfeladatról,” Mat. Fiz. Lapok 48, 436–452 (1941).MathSciNetzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  1. 1.University of ChicagoChicagoUSA
  2. 2.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

Personalised recommendations