On Bipartite Graphs with Linear Ramsey Numbers

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.

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Received December 1, 1999

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ID="*" Supported by NSF grant DMS-9704114

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ID="**" Supported by KBN grant 2 P03A 032 16

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Graham, R., Rödl, V. & Ruciński, A. On Bipartite Graphs with Linear Ramsey Numbers. Combinatorica 21, 199–209 (2001). https://doi.org/10.1007/s004930100018

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