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Polynomial Schur’s Theorem

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Abstract

We resolve the Ramsey problem for {x,y,z: x + y = p(z)} for all polynomials p over ℤ. In particular, we characterise all polynomials that are 2-Ramsey, that is, those p(z) such that any 2-colouring of ℕ contains infinitely many monochromatic solutions for x+y=p(z). For polynomials that are not 2-Ramsey, we characterise all 2-colourings of ℕ that are not 2-Ramsey, revealing that certain divisibility barrier is the only obstruction to 2-Ramseyness for x + y = p(z).

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Acknowledgements

We would like to thank two anonymous referees for their useful comments.

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Authors and Affiliations

  1. Extremal Combinatorics and Probability Group (ECOPRO), Institute for Basic Science (IBS), Daejeon, South Korea

    Hong Liu

  2. Institute of Mathematics, Budapest University of Technology and Economics, H-1529 B.O. Box, Budapest, Hungary

    Csaba Sándor

  3. Department of Computer Science and Information Theory, Budapest University of Technology and Economics, Műegyetem rkp. 3, H-1111, Budapest, Hungary

    Péter Pál Pach & Csaba Sándor

  4. MTA-BME Lendület Arithmetic Combinatorics Research Group, ELKH, Műegyetem rkp. 3, H-1111, Budapest, Hungary

    Péter Pál Pach & Csaba Sándor

  5. Department of Computer Science and DIMAP, University of Warwick, Coventry, CV4 7AL, UK

    Péter Pál Pach

Authors
  1. Hong Liu
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  2. Péter Pál Pach
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  3. Csaba Sándor
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Corresponding author

Correspondence to Péter Pál Pach.

Additional information

Hong Liu was supported by the Institute for Basic Science (IBS-R029-C4) and the UK Research and Innovation Future Leaders Fellowship MR/S016325/1.

Péter Pál Pach was partially supported by the National Research, Development and Innovation Office NKFIH (Grant Nr. PD115978 and K129335) and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. The author has also received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 648509). This publication reflects only its author’s view; the European Research Council Executive Agency is not responsible for any use that may be made of the information it contains.

Csaba Sándor was supported by the OTKA Grant No. K109789 and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.

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Liu, H., Pach, P.P. & Sándor, C. Polynomial Schur’s Theorem. Combinatorica 42 (Suppl 2), 1357–1384 (2022). https://doi.org/10.1007/s00493-021-4815-z

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  • DOI: https://doi.org/10.1007/s00493-021-4815-z

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