Advertisement

The Sub-Exponential Transition for the Chromatic Generalized Ramsey Numbers

Article

Abstract

A simple graph-product type construction shows that for all natural numbers rq, there exists an edge-coloring of the complete graph on 2 r vertices using r colors where the graph consisting of the union of any q color classes has chromatic number 2 q . We show that for each fixed natural number q, if there exists an edge-coloring of the complete graph on n vertices using r colors where the graph consisting of the union of any q color classes has chromatic number at most 2 q − 1, then n must be sub-exponential in r. This answers a question of Conlon, Fox, Lee, and Sudakov.

Mathematics Subject Classification (2010)

05C55 05D10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    D. Conlon, J. Fox, C. Lee and B. Sudakov: On the grid Ramsey problem and related questions, Int. Math. Res. Not..Google Scholar
  2. [2]
    D. Conlon, J. Fox, C. Lee and B. Sudakov: The Erdős-Gyárfás problem on generalized Ramsey numbers, Proc. London Math. Soc., to appear.Google Scholar
  3. [3]
    D. Eichhorn and D. Mubayi: Edge-coloring cliques with many colors on subcliques, Combinatorica 20 (2000), 441–444.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    P. Erdős: Problems and results on finite and infinite graphs, in: Recent advances in graph theory (Proc. Second Czechoslovak Sympos., Prague, 1974), 183–192, Academia, Prague, 1975.Google Scholar
  5. [5]
    P. Erdős: Solved and unsolved problems in combinatorics and combinatorial number theory, in: Proceedings of the Twelfth Southeastern Conference on Combinatorics, Graph Theory and Computing, Vol. I (Baton Rouge, La., 1981), Congr. Numer. 32 (1981), 49–62.MathSciNetMATHGoogle Scholar
  6. [6]
    P. Erdős and A. Gyárfás: A variant of the classical Ramsey problem, Combinatorica 17 (1997), 459–467.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    S. Gerke, Y. Kohayakawa, V. Rödl and A. Steger: Small subsets inherit sparse ε-regularity, J. Combin. Theory Ser. B 97 (2007), 34–56.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    D. Mubayi: Edge-coloring cliques with three colors on all 4-cliques, Combinatorica 18 (1998), 293–296.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    Y. Peng, V. Rödl and A. Ruciński: Holes in graphs, Electron. J. Combin. 9 (2002), Research Papers 1.Google Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsMITCambridgeUSA

Personalised recommendations