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Ramsey numbers of sparse hypergraphs
Institution:1. Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan;2. Taida Institute for Mathematical Sciences, National Taiwan University, Taipei 10617, Taiwan;3. National Center for Theoretical Sciences, Taipei Office, Taiwan;1. Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China;2. LaBRI UMR CNRS 5800, Université Bordeaux I, 33405 Talence Cedex, France;1. Institut de Mathématiques de Toulouse, Université Toulouse III, France;2. Laboratoire d''Informatique du Parallélisme, ENS Lyon, France;3. Laboratoire G-SCOP, CNRS, Université Grenoble-Alpes, Grenoble, France
Abstract:Chvátal, Rödl, Szemerédi and Trotter V. Chvátal, V. Rödl, E. Szemerédi and W.T. Trotter, The Ramsey number of a graph with a bounded maximum degree, J. Combinatorial Theory B 34 (1983), 239–243] proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. In O. Cooley, N. Fountoulakis, D. Kühn and D. Osthus, 3-uniform hypergraphs of bounded degree have linear Ramsey numbers, submitted] and B. Nagle, S. Olsen, V. Rödl and M. Schacht, On the Ramsey number of sparse 3-graphs, preprint] the same result was proved for 3-uniform hypergraphs. In O. Cooley, N. Fountoulakis, D. Kühn and D. Osthus, Embeddings and Ramsey numbers of sparse k-uniform hypergraphs, submitted] we extended this result to k-uniform hypergraphs for any integer k3. As in the 3-uniform case, the main new tool which we proved and used is an embedding lemma for k-uniform hypergraphs of bounded maximum degree into suitable k-uniform ‘quasi-random’ hypergraphs.
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