Abstract
For \(k \in \mathbb {N}\), write S(k) for the largest natural number such that there is a k-colouring of \(\{1, \ldots ,S(k)\}\) with no monochromatic solution to \(x-y=z^2\). That S(k) exists is a result of Bergelson, and a simple example shows that \(S(k) \ge 2^{2^{k-1}}\). The purpose of this note is to show that \(S(k) \le 2^{2^{2^{O(k)}}}\) .
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bergelson, V.: A density statement generalizing Schur's theorem. J. Combin. Theory Ser. A 43, 338–343 (1986)
V. Bergelson, Ergodic Ramsey theory—an update, in: Ergodic Theory of \(\bf Z\it ^d\) Actions (Warwick, 1993–1994), London Math. Soc. Lecture Note Ser., vol. 228, Cambridge Univ. Press (Cambridge, 1996), pp. 1–61
E. S. Croot and V. F. Lev, Open problems in additive combinatorics, in: Additive Combinatorics, CRM Proc. Lecture Notes, vol. 43, Amer. Math. Soc. (Providence, RI, 2007), pp. 207–233
Csikvári, P., Gyarmati, K., Sárközy, A.: Density and Ramsey type results on algebraic equations with restricted solution sets. Combinatorica 32, 425–449 (2012)
B. J. Green and S. Lindqvist, Monochromatic solutions to \(x + y = z^2\), Canadian J. Math. (2019), 1–27
Khalfalah, A., Szemerédi, E.: On the number of monochromatic solutions of \({x+y=z^2}\). Combin. Probab. Comput. 15, 213–227 (2006)
Lindqvist, S.: Partition regularity of generalised Fermat equations. Combinatorica 38, 1457–1483 (2018)
S. Lindqvist, Quadratic phenomena in additive combinatorics and number theory, DPhil thesis, University of Oxford (2019)
S. Prendiville, Counting monochromatic solutions to diagonal Diophantine equations, arXiv:2003.10161 (2020)
A. Sárkőzy, On difference sets of sequences of integers. I, Acta Math. Acad. Sci. Hungar., 31 (1978), 125–149
Acknowledgements
The author should like to thank the referee for a careful reading of the paper; the editors of the volume for the invitation to submit; and most importantly Endre Szemerédi for many years of support and interesting discussions.
Author information
Authors and Affiliations
Corresponding author
Additional information
To Endre Szemerédi on his 80th birthday
Rights and permissions
About this article
Cite this article
Sanders, T. On monochromatic solutions to \(x-y=z^2\). Acta Math. Hungar. 161, 550–556 (2020). https://doi.org/10.1007/s10474-020-01079-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-020-01079-6