, 31:39 | Cite as

On K s -free subgraphs in K s+k -free graphs and vertex Folkman numbers

  • Andrzej Dudek
  • Vojtěch Rödl


Extending the problem of determining Ramsey numbers Erdős and Rogers introduced the following function. For given integers 2 ≤ s < t let f s,t (n) = min{max{|S|: SV (H) and H[S] contains no K s }}, where the minimum is taken over all K t -free graphs H of order n. This function attracted a considerable amount of attention but despite that, the gap between the lower and upper bounds is still fairly wide. For example, when t=s+1, the best bounds have been of the form Ω(n 1/2+o(1)) ≤ f s,s+1(n) ≤ O(n 1−ɛ(s)), where ɛ(s) tends to zero as s tends to infinity. In this paper we improve the upper bound by showing that f s,s+1(n) ≤ O(n 2/3). Moreover, we show that for every ɛ > 0 and sufficiently large integers 1 ≪ ks, Ω(n 1/2−ɛ ) ≤ f s,s+k (n) ≤ O(n 1/2+ɛ . In addition, we also discuss some connections between the function f s,t and vertex Folkman numbers.

Mathematics Subject Classification (2000)

05C35 05C55 


  1. [1]
    N. Alon and M. Krivelevich: Constructive bounds for a Ramsey-type problem, Graphs Combin. 13 (1997), 217–225.MathSciNetzbMATHGoogle Scholar
  2. [2]
    B. Bollobás and H. R. Hind: Graphs without large triangle free subgraphs, Discrete Math. 87(2) (1991), 119–131.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    J. Brown and V. Rödl: A Ramsey type problem concerning vertex colourings, J. Combin. Theory Ser. B 52(1) (1991), 45–52.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    A. Dudek and V. Rödl: An almost quadratic bound on vertex Folkman numbers, J. Combin. Theory Ser. B 100(2) (2010), 132–140.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    A. Dudek and V. Rödl: New upper bound on vertex Folkman numbers, in: LATIN 2008 (E. Laber et al., eds.), Lecture Notes in Comput. Sci., vol. 4957, Springer, Berlin, 2008, pp. 473–478.CrossRefGoogle Scholar
  6. [6]
    N. Eaton and V. Rödl: A canonical Ramsey theorem, Random Structures Algorithms 3(4) (1992), 427–444.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    P. Erdős and C. A. Rogers: The construction of certain graphs, Canad. J. Math. 14 (1962), 702–707.MathSciNetCrossRefGoogle Scholar
  8. [8]
    J. Folkman: Graphs with monochromatic complete subgraphs in every edge coloring, SIAM J. Appl. Math. 18 (1970), 19–24.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    C. Godsil and G. Royle: Algebraic Graph Theory, Springer, New York, 2001.zbMATHCrossRefGoogle Scholar
  10. [10]
    N. Kolev and N. Nenov: New upper bound for a class of vertex Folkman numbers, Electron. J. Combin. 13 (2006), #R14.MathSciNetGoogle Scholar
  11. [11]
    M. Krivelevich: Bounding Ramsey numbers through large deviation inequalities, Random Structures Algorithms 7 (1995), 145–155.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    M. Krivelevich: K s-free graphs without large K r-free subgraphs, Combin. Probab. Comput. 3 (1994), 349–354.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    T. Łuczak, A. Ruciński and S. Urbański: On minimal vertex Folkman graphs, Discrete Math. 236 (2001), 245–262.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    N. Nenov: On the triangle vertex Folkman numbers, Discrete Math. 271 (2003), 327–334.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    J. Nešetřil and V. Rödl: The Ramsey property for graphs with forbidden complete subgraphs, J. Combin. Theory Ser. B 20(3) (1976), 243–249.zbMATHCrossRefGoogle Scholar
  16. [16]
    B. Sudakov: Large K r-free subgraphs in K s-free graphs and some other Ramseytype problems, Random Structures Algorithms 26 (2005), 253–265.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    B. Sudakov: A new lower bound for a Ramsey-type problem, Combinatorica 25(4) (2005), 487–498.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    J. Thas: Generalized polygons, in: F. Buekenhout (ed.), Handbook on Incidence Geometry, North-Holland, Amsterdam, 1995, pp. 383–431.CrossRefGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer Verlag 2011

Authors and Affiliations

  1. 1.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA
  2. 2.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA

Personalised recommendations