Skip to main content

Polynomial Schur’s Theorem

Combinatorica volume 42, pages 1357–1384 (2022)Cite this article

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).

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. V. Bergelson: Ergodic Ramsey theory. In Logic and combinatorics (Arcata, Calif., 1985), Contemp. Math. 65 (1987), 63–87, Amer. Math. Soc., Providence, RI, 1987.

    Article  Google Scholar 

  2. V. Bergelson and A. Leibman: Polynomial extensions of van der Waerden’s and Szemerédi’s theorems, J. Amer. Math. Soc. 9 (1996), 725–753.

    Article  MathSciNet  MATH  Google Scholar 

  3. S. Chow, S. Lindqvist and S. Prendiville: Rado’s criterion over squares and higher powers, J. Euro. Math. Soc., to appear.

  4. P. Csikvári, K. Gyarmati and A. Sárközy: Density and Ramsey type results on algebraic equations with restricted solution sets, Combinatorica 32 (2012), 425–449.

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Di Nasso and L. Luperi Baglini: Ramsey properties of nonlinear Diophantine equations, Advances in Mathematics 324 (2018), 84–117.

    Article  MathSciNet  MATH  Google Scholar 

  6. H. Furstenberg: Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. d’Analyse Math. 31 (1977), 204–256.

    Article  MathSciNet  MATH  Google Scholar 

  7. B. Green and S. Lindqvist: Monochromatic solutions to x + y = z2, Canadian Journal of Mathematics 71 (2019), 579–605.

    Article  MathSciNet  MATH  Google Scholar 

  8. B. Green and T. Sanders: Monochromatic sums and products, Discrete Analysis 5 (2016), 48.

    MathSciNet  MATH  Google Scholar 

  9. A. Khalafallah and E. Szemerédi: On the Number of Monochromatic Solutions of x + y = z2, Combinatorics, Probability and Computing 15 (2006), 213–227.

    Article  MathSciNet  Google Scholar 

  10. M. Kneser: Abschätzungen der asymptotischen Dichte von Summenmengen, Math. Zeitschr. 58 (1958), 459–484.

    Article  MATH  Google Scholar 

  11. S. Lindqvist: Partition regularity for generalised Fermat equations, Combinatorica, to appear.

  12. F. Mertens: Ein Beitrag zur analytischen Zahlentheorie, J. reine angew. Math. 78 (1874), 199–245.

    MathSciNet  Google Scholar 

  13. J. Moreira: Monochromatic sums and products in N, Annals of Mathematics 185 (2017), 1069–1090.

    Article  MathSciNet  MATH  Google Scholar 

  14. P. P. Pach: Monochromatic solutions to x + y = z2 in the interval [N, cN4], Bulletin of the London Mathematical Society 50 (2018), 1113–1116.

    Article  MathSciNet  MATH  Google Scholar 

  15. C. Pohoata: Boole’s formula as a consequence of Lagrange’s interpolation formula, Integers 8 (2008), A23.

    MathSciNet  MATH  Google Scholar 

  16. R. Rado: Studien zur Kombinatorik, Math. Z. 36 (1933), 424–470.

    Article  MathSciNet  MATH  Google Scholar 

  17. A. Robertson and D. Zeilberger: A 2-coloring of [1, N] can have (1/22)N2+O(N) monochromatic Schur triples, but not less!, Electronic Journal of Combinatorics 5 (1998), R19.

    Article  MATH  Google Scholar 

  18. A. Sárközy: On difference sets of sequences of integers. I, Acta Math. Acad. Sci. Hungar. 31 (1978), 125–149.

    Article  MathSciNet  MATH  Google Scholar 

  19. T. Schoen: The number of monochromatic Schur triples, Euro. J. Combinatorics 20 (1999), 855–866.

    Article  MathSciNet  MATH  Google Scholar 

  20. I. Schur: Über die Kongruenz xm + ym ≡ zm (mod p), Jahresber. Dtsch. Math.-Ver. 14 (1916), 114–117.

    Google Scholar 

  21. E. Szemerédi: On sets of integers containing no k elements in arithmetic progression, Acta Arithmetica 27 (1975), 199–245.

    Article  MathSciNet  MATH  Google Scholar 

  22. B. L. van der Waerden: Beweis einer baudetschen vermutung, Nieuw. Arch. Wisk. 15 (1927), 212–216.

    MATH  Google Scholar 

Download references

Acknowledgements

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

Author information

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
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Péter Pál Pach
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Csaba Sándor
    View author publications

    You can also search for this author in PubMed Google Scholar

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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

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

Mathematics Subject Classification (2010)

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