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 Burr 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 there exists a bipartite graph with n vertices and maximum degree at most Delta whose ramsey number is greater than c(Delta)n for some absolute constant c > 1.