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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