Graphs and Combinatorics

, Volume 2, Issue 1, pp 135–144 | Cite as

Large triangle-free subgraphs in graphs withoutK4

  • P. Frankl
  • V. Rödl


It is shown that for arbitrary positiveε there exists a graph withoutK4 and so that all its subgraphs containing more than 1/2 +ε portion of its edges contain a triangle (Theorem 2). This solves a problem of Erdös and Nešetřil. On the other hand it is proved that such graphs have necessarily low edge density (Theorem 4).

Theorem 3 which is needed for the proof of Theorem 2 is an analog of Goodman's theorem [8], it shows that random graphs behave in some respect as sparse complete graphs.

Theorem 5 shows the existence of a graph on less than 1012 vertices, withoutK4 and which is edge-Ramsey for triangles.


Random Graph Complete Graph Edge Density 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alon, N.: Personal communicationGoogle Scholar
  2. 2.
    Bollobás, B.: Extermal Graph Theory. London-New York-San Francisco: Academic Press 1978Google Scholar
  3. 3.
    Chernoff, H.: A measure of asymptotic efficiency for tests of hypothesis based on the sum of observations. Ann. Stat.23, 493–502 (1952)Google Scholar
  4. 4.
    Erdös, P.: Problems and results on finite and infinite graphs. In: Recent Advances in Graph Theory (Proceedings of the Symposium held in Prague 1974) edited by M. Fiedler, pp 183–192. Praha: Academia 1974Google Scholar
  5. 5.
    Erdös, P.: On some of my conjectures in number theory and combinatorics. Congr. Numerantium39, 3–19 (1983)Google Scholar
  6. 6.
    Erdös, P., Spencer, J.: Probabilistic Methods in Combinatorics. Budapest: Akadémiai Kiado 1974Google Scholar
  7. 7.
    Folkman, J.: Graphs with monochromatic complete subgraphs in every edge colouring. SIAM J. Appl. Math.18, 19–29 (1970)Google Scholar
  8. 8.
    Goodman, A.W.: On sets of acquaintances and strangers at any party. Amer. Math. Mon.66, 778–783 (1959)Google Scholar
  9. 9.
    Graham, R.L.: On edgewise 2-coloured graphs with monochromatic triangles and containing no complete hexagon. J. Comb. Theory4, 300 (1968)Google Scholar
  10. 10.
    Graham, R.L.: Rudiments of Ramsey theory. Reg. Conf. Ser. Math.45 (1981)Google Scholar
  11. 11.
    Irving, R.W.: On a bound of Graham and Spencer for a graph colouring constant. J. Comb. Theory (B)15, 200–203 (1973)Google Scholar
  12. 12.
    Nešetřil, J., Rödl, V.: The Ramsey property for graphs with forbidden complete subgraphs. J. Comb. Theory (B)20, 243–249 (1976)Google Scholar
  13. 13.
    Nešetřil, J., Rödl, V.: Partition theory and its applications. Lond. Math. Soc. Lect. Note Ser.38, 96–157 (1979)Google Scholar
  14. 14.
    Rödl, V.: On universality of graphs with uniformly distributed edges. Discrete Math. (to appear)Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • P. Frankl
    • 1
  • V. Rödl
    • 2
    • 3
  1. 1.CNRSQuai Anatole FranceParisFrance
  2. 2.Department of MathematicsFJFI, ČVUTPraha 1Czechoslovakia
  3. 3.AT&T Bell LaboratoriesMurray HillUSA

Personalised recommendations