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Monochromatic Solutions to $x+y=z^{2}$

Part of: Extremal combinatorics Sequences and sets

Published online by Cambridge University Press:  07 January 2019

Ben Joseph Green  and
Sofia Lindqvist
Ben Joseph Green
Affiliation:
Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Rd, Oxford OX2 6GG Email: ben.green@maths.ox.ac.uklindqvist.sofia@gmail.com
Sofia Lindqvist
Affiliation:
Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Rd, Oxford OX2 6GG Email: ben.green@maths.ox.ac.uklindqvist.sofia@gmail.com
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Abstract

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Suppose that N is 2-coloured. Then there are infinitely many monochromatic solutions to $x+y=z^{2}$. On the other hand, there is a 3-colouring of N with only finitely many monochromatic solutions to this equation.

Keywords

additive combinatoricsRamsey theory

MSC classification

Primary: 11B75: Other combinatorial number theory
Secondary: 05D10: Ramsey theory
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 3 , June 2019 , pp. 579 - 605
Copyright
© Canadian Mathematical Society 2018 

Footnotes

This work was partially supported by a grant from the Simons Foundation (award number 376201 to Ben Green) , and the first author is supported by ERC Advanced Grant AAS 279438. We thank both organisations for their support.

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