  We give a short proof that any k-uniform hypergraph H on n vertices with
bounded degree \Delta has Ramsey number at most c(\Delta, k)n, for an
appropriate constant c(\Delta, k). This result was recently proved by several
authors, but those proofs are all based on applications of the hypergraph
regularity method. Here we give a much simpler, self-contained proof which uses
new techniques developed recently by the authors together with an argument of
Kostochka and R\"odl. Moreover, our method demonstrates that, for k \geq 4,
c(\Delta, k) \leq 2^{2^{\Ddots^{2^{c \Delta}}}}, where the tower is of height k
and the constant c depends on k. It significantly improves on the
Ackermann-type upper bound that arises from the regularity proofs, and we
present a construction which shows that, at least in certain cases, this bound
is not far from best possible. Our methods also allows us to prove quite sharp
results on the Ramsey number of hypergraphs with at most m edges.
