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Polynomial Schur’s Theorem


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

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Correspondence to Péter Pál Pach.

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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).

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Mathematics Subject Classification (2010)

  • 05D10
  • 11A07
  • 11B30
  • 11T06
  • 11B75