On bipartite graphs with linear Ramsey numbers

We provide an elementary proof of the fact that the ramsey number of every bipartite graph H with maximum degree at most Delta is less than 8(8 Delta) (Delta)\V(H)\. 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 B urr and Erdos. Applying the probabilistic method we also show that for all Delta greater than or equal to 1 and n greater than or equal to Delta + 1 t here exists a bipartite graph with n vertices and maximum degree at most De lta whose ramsey number is greater than c(Delta)n for some absolute constan t c > 1.