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A Note on Induced Ramsey Numbers

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A Journey Through Discrete Mathematics

Abstract

The induced Ramsey number r ind(F) of a k-uniform hypergraph F is the smallest natural number n for which there exists a k-uniform hypergraph G on n vertices such that every two-coloring of the edges of G contains an induced monochromatic copy of F. We study this function, showing that r ind(F) is bounded above by a reasonable power of r(F). In particular, our result implies that \(r_{\mathrm{ind}}(F) \leq 2^{2^{ct}}\) for any 3-uniform hypergraph F with t vertices, mirroring the best known bound for the usual Ramsey number. The proof relies on an application of the hypergraph container method.

The first author was supported by a Royal Society University Research Fellowship.

The fourth author was partially supported by NSF grants DMS-1102086 and DMS-1301698.

The fifth author was supported through the Heisenberg-Programme of the DFG.

Dedicated to the memory of Jirka Matoušek

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Oxford, UK

    David Conlon

  2. Department of Mathematics and Computer Science, Emory University, Atlanta, GA, USA

    Domingos Dellamonica, Steven La Fleur & Vojtěch Rödl

  3. Fachbereich Mathematik, Universität Hamburg, Hamburg, Germany

    Mathias Schacht

Authors
  1. David Conlon
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  2. Domingos Dellamonica
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  3. Steven La Fleur
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  4. Vojtěch Rödl
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  5. Mathias Schacht
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Correspondence to Vojtěch Rödl .

Editor information

Editors and Affiliations

  1. Department of Applied Mathematics, Charles University, Praha, Czech Republic

    Martin Loebl

  2. Computer Science Institute of Charles University, Charles University, Praha, Czech Republic

    Jaroslav Nešetřil

  3. School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, USA

    Robin Thomas

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Conlon, D., Dellamonica, D., La Fleur, S., Rödl, V., Schacht, M. (2017). A Note on Induced Ramsey Numbers. In: Loebl, M., Nešetřil, J., Thomas, R. (eds) A Journey Through Discrete Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-44479-6_13

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