Embeddings and Ramsey numbers of sparse κ-uniform hypergraphs

Abstract

Chvátal, Rödl, Szemerédi 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.

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Correspondence to Deryk Osthus.

Additional information

N. Fountoulakis and D. Kühn were supported by EPSRC, grant no. EP/D50564X/1.

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Cooley, O., Fountoulakis, N., Kühn, D. et al. Embeddings and Ramsey numbers of sparse κ-uniform hypergraphs. Combinatorica 29, 263–297 (2009). https://doi.org/10.1007/s00493-009-2356-y

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Mathematics Subject Classification (2000)

  • 05D10
  • 05C65