Embeddings and Ramsey numbers of sparse k-uniform hypergraphs

Chvatal, Rodl, Szemer,di and Trotter [3] proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. In [6,23] the same result was proved for 3-uniform hypergraphs. Here we extend this result to kappa-uniform hypergraphs for any integer kappa a parts per thousand yen 3. As in the 3-uniform case, the main new tool which we prove and use is an embedding lemma for kappa-uniform hypergraphs of bounded maximum degree into suitable kappa-uniform 'quasi-random' hypergraphs.