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Big Ramsey Degrees of 3-Uniform Hypergraphs Are Finite

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Abstract

We prove that the universal homogeneous 3-uniform hypergraph has finite big Ramsey degrees. This is the first case where big Ramsey degrees are known to be finite for structures in a non-binary language.

Our proof is based on the vector (or product) form of Milliken’s Tree Theorem and demonstrates a general method to carry existing results on structures in binary relational languages to higher arities.

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Acknowledgement

We would like to thank Stevo Todorcevic for his kind remarks and for helpful discussion regarding the history and context of this area. We are also grateful to three anonymous referees whose suggestions improved the presentation of this paper.

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Authors and Affiliations

  1. Department of Applied Mathematics (KAM), Charles University, Malostranské náměstí 25, Praha 1, Czech Republic

    Martin Balko, David Chodounský, Jan Hubička & Matěj Konečný

  2. Department of Mathematics, Universitat Politècnica de Catalunya, Jordi Girona 1-3, Barcelona, Spain

    Lluis Vena

Authors
  1. Martin Balko
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  2. David Chodounský
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  3. Jan Hubička
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  4. Matěj Konečný
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  5. Lluis Vena
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Corresponding author

Correspondence to Jan Hubička.

Additional information

All authors are supported by project 18-13685Y of the Czech Science Foundation (GAČR). The first author is additionally supported by Center for Foundations of Modern Computer Science (Charles University project UNCE/SCI/004) and by the PRIMUS/17/SCI/3 project of Charles University. Fourth author is supported by the Charles University project GA UK No 378119. Fifth author is supported by Beatriu de Pinós BP2018, funded by the AGAUR (Government of Catalonia) and by the Horizon 2020 programme No 801370. This paper is part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 810115).

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Balko, M., Chodounský, D., Hubička, J. et al. Big Ramsey Degrees of 3-Uniform Hypergraphs Are Finite. Combinatorica 42, 659–672 (2022). https://doi.org/10.1007/s00493-021-4664-9

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  • DOI: https://doi.org/10.1007/s00493-021-4664-9

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