The Sub-Exponential Transition for the Chromatic Generalized Ramsey Numbers

Abstract

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

This is a preview of subscription content, access via your institution.

References

  1. [1]

    D. Conlon, J. Fox, C. Lee and B. Sudakov: On the grid Ramsey problem and related questions, Int. Math. Res. Not..

  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.

  3. [3]

    D. Eichhorn and D. Mubayi: Edge-coloring cliques with many colors on subcliques, Combinatorica 20 (2000), 441–444.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  MATH  Google Scholar 

  6. [6]

    P. Erdős and A. Gyárfás: A variant of the classical Ramsey problem, Combinatorica 17 (1997), 459–467.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google Scholar 

  8. [8]

    D. Mubayi: Edge-coloring cliques with three colors on all 4-cliques, Combinatorica 18 (1998), 293–296.

    MathSciNet  Article  MATH  Google Scholar 

  9. [9]

    Y. Peng, V. Rödl and A. Ruciński: Holes in graphs, Electron. J. Combin. 9 (2002), Research Papers 1.

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Choongbum Lee.

Additional information

Research supported by NSF Grant DMS-1362326.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lee, C., Tran, B. The Sub-Exponential Transition for the Chromatic Generalized Ramsey Numbers. Combinatorica 39, 355–376 (2019). https://doi.org/10.1007/s00493-017-3474-6

Download citation

Mathematics Subject Classification (2010)

  • 05C55
  • 05D10