Green 2019 monochromatic solutions x plus y z squared

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theorem_1_1: Green and Lindqvist's main theorem: some 3-coloring of N has no monochromatic solution of x + y = z^2 besides the trivial x = y = z = 2, while every 2-coloring of N has infinitely many monochromatic solutions, with x, y and z all of one color.

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Ben Joseph Green and Sofia Lindqvist, Monochromatic solutions to $x+y=z^2$, Canad. J. Math. 71 (2019), no. 3, 579--605, DOI 10.4153/CJM-2017-036-1; received 29 August 2017, published electronically 17 January 2018 (p. 579); arXiv:1608.08374 (per Sanders's reference [GL19]). The Canadian Journal of Mathematics is a refereed journal. Not a source key of the site; Problem 439's page cites it as [GrLi19].

Copy read. The copy read for this card is the journal's typeset article: twenty-seven pages carrying the journal's pagination 579--605 (printed page $p$ is PDF page $p-578$), every foot reading "https://doi.org/10.4153/CJM-2017-036-1 Published online by Cambridge University Press", with a complete text layer. Provenance: 1,274,470 bytes, from the repository's survey download set of September 2026 (the retrieval date and URL of the set were not recorded; the DOI above is the article's public address). The survey set filed the copy under Pach's name, misattributing the paper to Pach, whose 2018 paper on monochromatic solutions of $x+y=z^2$ in $[N,cN^4]$ is a different paper. The file prints "©Canadian Mathematical Society 2018" in the header of its first page, every other right reserved.

Read status: claims checked for the abstract, Theorem 1.1 and the outline of the proof (p. 579), and the remarks on the $[N,CN^8]$ strengthening, on Khalfalah and Szemerédi [9] and on the modular version (p. 580), read clause by clause in the text layer and, for p. 580, on the page image; the proof (Sections 2--7 and Appendix A, pp. 580--604) was not checked: Section 2 (pp. 580--581) and the end of Section 7 (pp. 598--600) were read for the proof pointer on the result page, the rest only for the statements it cites; the reference list (pp. 604--605) was read for [7], [9] and [12]. Result page: theorem_1_1.

Contents

• Theorem 1.1 (p. 579): "There is a 3-colouring of $\mathbb N$ with no monochromatic solution to $x+y=z^2$ other than the trivial one. On the other hand, every 2-colouring of $\mathbb N$ has infinitely many monochromatic solutions to $x+y=z^2$." The introduction records that Csikvári, Gyarmati and Sárközy (their [7], listed as K. Gyarmati, P. Csikvári and A. Sárközy, Combinatorica 32 (2012), 425--449) showed the equation is not partition regular with a 16-coloring whose only monochromatic solution is $x=y=z=2$; the theorem settles the optimal number of colors.
• The outline (pp. 579--580): the 3-coloring is elementary (Section 2); the 2-coloring statement uses the arithmetic regularity lemma, Fourier and Diophantine arguments, a result of Lagarias, Odlyzko and Shearer, and gaps between constrained sums of two squares, to show that if neither color class contained a solution, one class would contain all large multiples of some $q$, and with them a solution.
• Remarks (p. 580): the paper says its arguments in fact show that, once $N$ is large, every 2-coloring of $[N,CN^8]$ contains a monochromatic solution, with $C$ an absolute but astronomically large constant, and leaves this to the reader, the proof as written not yielding it directly; the paper then points to Khalfallah and Szemerédi [9], whose title is similar but whose problem is different, and states their result (quoted): "They show that any finite colouring of $\mathbb N$ contains a solution to $x+y=z^2$ with $x$ and $y$ having the same colour (but not necessarily $z$)." Reference [9] (p. 605): A. Khalfallah and E. Szemerédi, On the number of monochromatic solutions of $x+y=z^2$, Combin. Probab. Comput. 15 (2006), no. 1--2, 213--227 (the site's KhSz06; the paper spells the first author "Khalfallah"). For primes $p>p_0(k)$ the second author (their [12], listed as S. Lindqvist, Partition regularity of generalised Fermat equations, arXiv:1606.07334) found $\gg_k p^2$ monochromatic solutions of $x+y=z^2$ in every $k$-coloring of $\mathbb Z/p\mathbb Z$.

Compiled scope

Statements at claims-checked depth for pp. 579--580; no proof was checked and nothing here is independently reviewed. The Khalfalah--Szemerédi paper is not held; its theorem is consumed through this remark and through Sanders's introduction, filed as sanders_2020_monochromatic_solutions_x_minus_y_z_squared.

Bears on. #439: the remark on p. 580 (= PDF p. 2, page image) attests the Khalfalah--Szemerédi theorem, a solution of $x+y=z^2$ with $x$ and $y$ of one color and $z$ unconstrained, though without the condition $x\ne y$ that the problem asks for, and distinguishes it from the fully monochromatic question; Theorem 1.1 (p. 579) settles that fully monochromatic question for squares (two colors force infinitely many solutions, three do not), an adjacent result the site does not ask, and as printed it does not state $x\ne y$ for the solutions it supplies; context, not a source of the status. The relation is stated on Theorem 1.1's page.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.