Consider the following one player game on an empty graph with n vertices. The edges are presented one by one to the player in a random order. One of two colors, red or blue, has to be assigned to each edge immediately. The player’s object is to color as many edges as possible without creating a monochromatic clique K of some fixed size ℓ. We prove a threshold phenomenon for the expected duration of this game. We show that there is a function N 0 = N 0(ℓ, n) such that the player can asymptotically almost surely survive up to N(n) ≪ N 0 edges by playing greedily and that this is best possible, i.e., there is no strategy such that the game would last for N(n) ≫ N 0 edges.


Random Graph Graph Property Edge Coloring Base Graph Greedy Strategy 
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.
    Ramsey, F.P.: On a problem of formal logic. Proceedings of the London Mathematical Society 30, 264–286 (1930)CrossRefGoogle Scholar
  2. 2.
    Folkman, J.: Graphs with monochromatic complete subgraphs in every edge coloring. SIAM J. Appl. Math. 18, 19–24 (1970)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Nešetřil, J., Rödl, V.: The Ramsey property for graphs with forbidden complete subgraphs. J. Combinatorial Theory Ser. B 20, 243–249 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Rödl, V., Ruciński, A.: Threshold functions for Ramsey properties. J. Amer. Math. Soc. 8, 917–942 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Friedgut, E., Kohayakawa, Y., Rödl, V., Ruciński, A., Tetali, P.: Ramsey games against a one-armed bandit. Combinatorics, Probability and Computing 12, 515–545 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bollobás, B.: Threshold functions for small subgraphs. Math. Proc. Cambridge Philos. Soc. 90, 197–206 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Erdös, P., Rényi, A.: On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. 5, 17–61 (1960)zbMATHGoogle Scholar
  8. 8.
    Friedgut, E., Kalai, G.: Every monotone graph property has a sharp threshold. Proc. Amer. Math. Soc. 124, 2993–3002 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Friedgut, E.: Sharp thresholds of graph properties, and the k-sat problem. J. Amer. Math. Soc. 12, 1017–1054 (1999); With an appendix by Jean BourgainzbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Friedgut, E., Krivelevich, M.: Sharp thresholds for certain Ramsey properties of random graphs. Random Structures & Algorithms 17, 1–19 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Friedgut, E.: Hunting for sharp thresholds. Random Structures & Algorithms 26, 37–51 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Friedgut, E., Rödl, V., Ruciński, A., Tetali, P.: A sharp threshold for random graphs with monochromatic triangle in every edge coloring. Memoirs of the AMS (to appear)Google Scholar
  13. 13.
    Janson, S., Łuczak, T., Ruciński, A.: Random graphs. Wiley-Interscience, New York (2000)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Martin Marciniszyn
    • 1
  • Reto Spöhel
    • 1
  • Angelika Steger
    • 1
  1. 1.Institute of Theoretical Computer ScienceETH ZürichZürichSwitzerland

Personalised recommendations