---

Statement

Setting (pp. 1--2). $\mathbb E^n$ and $\ell_m$ are as in Theorem 1. For $\alpha\in\mathbb R_+$, $\alpha\ell_m$ is a set of $m$ points on a line with consecutive points at distance $\alpha$. The paper observes (p. 2) that $\mathbb E^n\to(\alpha_{red}\ell_{m_1},\alpha_{blue}\ell_{m_2})$ is equivalent to $\mathbb E^n\to(\ell_{m_1},(\alpha_{blue}/\alpha_{red})\ell_{m_2})$.

Theorem 2 (p. 2, quoted). "For any $n>0$, there exists a red/blue-coloring of $\mathbb E^n$ that does not contain any red copy of $\ell_3$ and any blue copy of $\alpha\ell_{8649}$, whenever $\alpha\in\mathbb R_+$ satisfies at least one of the following conditions:

• $\alpha^2\notin\mathbb Q$,
• $\alpha^2=p/q$, $p,q\in\mathbb N$ and $47\nmid q$,
• $\alpha^2\ge2$,
• $\alpha^2\le1/(7\cdot47^4\cdot48)$."

So $\mathbb E^n\not\to(\ell_3,\alpha\ell_{8649})$ for every $n>0$ and every such $\alpha$. The second condition is read as printed: some representation $p/q$ of $\alpha^2$ with $47\nmid q$. Here $8649=93^2$. For $\alpha=1$ the second condition holds, but Theorem 1 gives the shorter $\ell_{1177}$.

In its closing remarks (p. 12) the paper states that the authors believe $8649$ is far from optimal and that the conditions on $\alpha$ can be dropped, and that using different primes in the proof, the Pólya--Vinogradov inequality gives a finite bound for every $\alpha$; no such bound is stated or proved in the paper.

Source. Jakob Führer and Géza Tóth, Progressions in Euclidean Ramsey theory, European Journal of Combinatorics 125 (2025), 104105, doi:10.1016/j.ejc.2024.104105, arXiv:2402.12567: the statement on p. 2, the proof in Section 3 (pp. 6--12), the remarks on p. 12. Labels and pages are those of arXiv:2402.12567v1, the edition named on the source card.

Read depth. Claims checked: the statement and the statements of Lemmas 6--8 were read clause by clause on the printed pages. The proof was read for structure only. Nothing here is independently reviewed.

Proof pointer

Section 3, pp. 6--12. The coloring (p. 6) colors $x$ red when $\lfloor|x|^2\rfloor\in\{0,5,10,15,20\}+47\mathbb Z$, the analogue of the coloring of Theorem 1 with the prime $47$, and is applied to scaled red and blue progressions $\alpha_{red}\ell_3$ and $\alpha_{blue}\ell_{8649}$ with $\alpha=\alpha_{blue}/\alpha_{red}$. Lemma 6 (p. 6): if $x,y,z$ form a copy of $\alpha_{red}\ell_3$ with $47N+1\le\alpha_{red}^2\le47N+3/2$ and $N\in\mathbb Z_{\ge0}$, then $\lfloor|x|^2\rfloor-2\lfloor|y|^2\rfloor+\lfloor|z|^2\rfloor\in\{1,2,3,4\}$ modulo $47$. No three floors in $\{0,5,10,15,20\}+47\mathbb Z$ meet this, a check the paper leaves implicit. Lemma 7 (p. 7) states that no shift of the squares, or of the nonsquares together with $0$, of $\mathbb F_{47}$ avoids $\{0,5,10,15,20\}$. For $\alpha_{blue}^2=b+\epsilon_2$ with $b\in\mathbb N\setminus47\mathbb Z$ and $0<\epsilon_2<1/(7\cdot47^4\cdot48)$ (p. 7, where the bound is printed "$17(7\cdot47^4\cdot48)$" [sic]), Dirichlet's theorem with $N=47$ and Lemma 8 (pp. 7--10) show that the floors of the squared norms along every $d$-th point of a copy of $\alpha_{blue}\ell_{8649}$ cover such a shift modulo $47$, so one point is red. Sections 3.1--3.4 (pp. 11--12) then choose $\alpha_{red}$ and $\alpha_{blue}$ for each of the four conditions: $\alpha^2\ge2$, small $\alpha^2$, rational $\alpha^2$ (two cases), and irrational $\alpha^2$ by equidistribution, cited from Kuipers and Niederreiter. The opening of Section 3 (p. 6) speaks of "6628 blue points $x_0,x_1,...,x_{6627}$" [sic], while the argument that follows uses $8649$ points.

Dependencies

Lemma 1 of the same paper (see Theorem 1); Dirichlet's approximation theorem, cited from Schmidt; equidistribution of $(k\theta)$ for irrational $\theta$, cited from Kuipers and Niederreiter. No corpus result.

Bears on

• Problem 188: the problem forbids a red unit pair and a blue unit-step $k$-term progression. Theorem 2 forbids a red $\ell_3$, not a red unit pair, and at $\alpha=1$ it is weaker than Theorem 1, so it gives no bound on the problem's $k$.