Sanders 2020 monochromatic solutions x minus y z squared

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theorem_1_1: Sanders's main theorem: if some k-coloring of {1, ..., N} has no monochromatic solution of x - y = z^2, then N is at most a triple exponential in O(k); for k at least 2, a coloring with k classes on N = 2^(2^(k-1)) shows that the bound cannot drop below 2^(2^(k-1)).

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Tom Sanders, On monochromatic solutions to $x-y=z^2$, Acta Math. Hungar. 161 (2020), no. 2, 550--556, DOI 10.1007/s10474-020-01079-6; arXiv:2008.07297v1 (Crossref and arXiv records read, as recorded on Problem 439's page; the journal text was not compared). Acta Mathematica Hungarica is a refereed journal; the note is dedicated "To Endre Szemerédi on his 80th birthday" (p. 1) and thanks "the referee for a careful reading of the paper" and "the editors of the volume for the invitation to submit" (p. 6). Not a source key of the site; Problem 439's page cites it as [Sa20].

Edition read. The copy read for this card is the author's typescript: six pages numbered 1--6 (the journal's 550--556 do not appear), no arXiv stamp, with a complete text layer; its metadata dates it May 2020. Provenance: 310,410 bytes, downloaded in September 2026 (the retrieval date and URL were not recorded; the arXiv abstract page https://arxiv.org/abs/2008.07297 is a public address of the text). The downloaded copy was named with the year 2018, which misdates the paper; the journal year is 2020. The file carries no arXiv stamp and prints no notice; it is the author's own typescript, built in May 2020, three months before the arXiv submission of 17 August 2020, and the journal's version of record was not compared; the arXiv abstract page (https://arxiv.org/abs/2008.07297v1, read 2026-10-02) names arXiv's non-exclusive distribution license for the article, every other right reserved.

Read status: claims checked for the abstract, the introduction's account of Khalfalah--Szemerédi, of Csikvári--Gyarmati--Sárközy and Green--Lindqvist, of the Furstenberg--Sárközy theorem and Bergelson's coloring result, Theorem 1.1 and the lower-bound coloring with its verification (pp. 1--2), read clause by clause in the text layer and on the page images; the proof (Sections 2--3, pp. 2--6) was read for the proof pointer on the result page but not checked step by step; the reference list (p. 6) was read for [Ber86], [Ber96], [CGS12], [GL19], [KS06] and [Lin19].

Contents

• Abstract and Theorem 1.1 (pp. 1--2): with $S(k)$ the largest $N$ such that some $k$-coloring of $\{1,\ldots,N\}$ has no monochromatic solution to $x-y=z^2$ (which exists by Bergelson), $2^{2^{k-1}}\le S(k)\le2^{2^{2^{O(k)}}}$; the lower bound from the coloring with classes $\{1\}$ and $\{2^{2^i},\ldots,2^{2^{i+1}}\}$, $0\le i\le k-2$, verified on p. 2 (these cover $[N]$ only for $k\ge2$; $S(1)=1$, as the result page notes); Lindqvist's thesis gave an earlier quantitative bound. Result page: theorem_1_1.
• The introduction (p. 1) opens with the Khalfalah--Szemerédi theorem [KS06], presented as the answer to a question posed by Roth, Erdős, Sárközy and Sós: for every $r\in\mathbb N$ and every $N$ sufficiently large in terms of $r$, every $r$-coloring of $[N]:=\{1,\ldots,N\}$ contains, in the paper's words, "two distinct elements $x$ and $y$ with the same colour and $x+y=z^2$ for some natural $z$." Then: Csikvári, Gyarmati and Sárközy [CGS12, Theorem 3] showed $z$ cannot be required to share the color; Green and Lindqvist [GL19] refined this to 3-colorings without solutions with $x$, $y$ distinct and $x$, $y$, $z$ of one color, and showed 3 cannot be reduced to 2; for $x-y$ the Furstenberg--Sárközy theorem gives a density analog, and Bergelson [Ber96, p. 53] showed every $k$-coloring of $\mathbb N$ has $x-y=z^2$ with $x$, $y$, $z$ of one color.
• References (p. 6): [KS06] A. Khalfalah 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); [GL19] Green and Lindqvist, Canadian Journal of Mathematics, 2019, arXiv:1608.08374 (filed as green_2019_monochromatic_solutions_x_plus_y_z_squared); [CGS12] Csikvári, Gyarmati and Sárközy, Combinatorica 32 (2012), 425--449.

Compiled scope

Statements at claims-checked depth for pp. 1--2; the proof was read for a pointer but not checked, and nothing here is independently reviewed. The Khalfalah--Szemerédi paper is not held; its theorem for squares is consumed through this introduction's restatement, which is second-hand.

Bears on. #439: the introduction (p. 1 = PDF p. 1, page image) restates the Khalfalah--Szemerédi theorem for squares with the clause the problem needs, "two distinct elements $x$ and $y$ with the same colour and $x+y=z^2$", in the finite form on $[N]$ for every number of colors $r$, which implies the infinite form the site states; it attributes the question to Roth, Erdős, Sárközy and Sós; the $k$th-power case is not mentioned. Theorem 1.1 concerns the fully monochromatic equation $x-y=z^2$, an adjacent result that neither proves nor refutes any part of the problem; 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.