Embeddings and Ramsey numbers of sparse κ-uniform hypergraphs

Abstract

Chvatal, Rodl, Szemeredi 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 κ-uniform hypergraphs for any integer κ ≥ 3. As in the 3-uniform case, the main new tool which we prove and use is an embedding lemma for κ-uniform hypergraphs of bounded maximum degree into suitable κ-uniform ‘quasi-random’ hypergraphs.