  The Ramsey number r(H) of a graph H is the smallest number n such that, in
any two-colouring of the edges of K_n, there is a monochromatic copy of H. We
study the Ramsey number of graphs H with t vertices and density \r, proving
that r(H) \leq 2^{c \sqrt{\r} \log (2/\r) t}. We also investigate some related
problems, such as the Ramsey number of graphs with t vertices and maximum
degree \r t and the Ramsey number of random graphs in \mathcal{G}(t, \r), that
is, graphs on t vertices where each edge has been chosen independently with
probability \r.
