, Volume 21, Issue 2, pp 199–209

On Bipartite Graphs with Linear Ramsey Numbers

  • R. L. Graham
  • V. Rödl
  • A. Ruciński
Original Paper

DOI: 10.1007/s004930100018

Cite this article as:
Graham, R., Rödl, V. & Ruciński, A. Combinatorica (2001) 21: 199. doi:10.1007/s004930100018

Dedicated to the memory of Paul Erdős

We provide an elementary proof of the fact that the ramsey number of every bipartite graph H with maximum degree at most \(\) is less than \(\). This improves an old upper bound on the ramsey number of the n-cube due to Beck, and brings us closer toward the bound conjectured by Burr and Erdős. Applying the probabilistic method we also show that for all \(\) and \(\) there exists a bipartite graph with n vertices and maximum degree at most \(\) whose ramsey number is greater than \(\) for some absolute constant c>1.

AMS Subject Classification (2000) Classes:  05C55, 05D40, 05C80

Copyright information

© János Bolyai Mathematical Society, 2001

Authors and Affiliations

  • R. L. Graham
    • 1
  • V. Rödl
    • 2
  • A. Ruciński
    • 3
  1. 1.UCSD La Jolla; CA, USA; E-mail: graham@ucsd.eduUS
  2. 2.Emory University Atlanta; GA, USA; E-mail: rodl@mathcs.emory.eduUS
  3. 3.A. Mickiewicz University; Poznań, Poland; E-mail: