3-Uniform hypergraphs of bounded degree have linear Ramsey numbers

Chvátal, Rödl, Szemerédi and Trotter [V. Chvátal, V. Rödl, E. Szemerédi, W.T. Trotter Jr., The Ramsey number of a graph with a bounded maximum degree, J. Combin. Theory Ser. B 34 (1983) 239–243] proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. We prove that the same holds for 3-uniform hypergraphs. The main new tool which we prove and use is an embedding lemma for 3-uniform hypergraphs of bounded maximum degree into suitable 3-uniform ‘pseudo-random’ hypergraphs.