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.