Fuhrer 2025 progressions euclidean ramsey theory

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theorem_1: Führer and Tóth's explicit spherical red-blue coloring of Euclidean space, in every dimension, with no red three-term and no blue 1177-term collinear progression of unit spacing.

theorem_2: Führer and Tóth's red-blue colorings of Euclidean space, in every dimension, with no red unit three-term progression and no blue 8649-term progression of spacing alpha, whenever alpha^2 is irrational, a fraction whose denominator 47 does not divide, at least 2, or at most 1/(7 47^4 48).

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Jakob Führer, 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 arXiv record (https://arxiv.org/abs/2402.12567, read 2026-10-07) names the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 license. The copy read for this card is arXiv:2402.12567v1 (19 February 2024); the journal text was not compared.

Writing ell_m for m collinear points at consecutive distance 1, Theorem 1 (p. 2) constructs, for every n > 0, a red/blue coloring of E^n containing no red copy of ell_3 and no blue copy of ell_1177, improving the bound m = 10^50 of Conlon and Wu. The coloring (p. 3) is explicit and spherical, depending only on the squared distance from the origin: red is {x : floor(|x|^2) in {0,4,8,12} + 29Z}. Lemma 1 (p. 2) records that a copy of alpha ell_3 satisfies |x|^2 - 2|y|^2 + |z|^2 = 2 alpha^2, Lemma 2 (p. 3) turns this into X - 2Y + Z in {1,2,3} for the integer parts of the squared norms of a copy of ell_3, and the paper checks that this has no solution with X, Y, Z in {0,4,8,12} modulo 29, so there is no red ell_3. Theorem 2 (p. 2) handles a different blue step: for every n > 0 there is a coloring of E^n with no red ell_3 and no blue copy of alpha ell_8649, whenever alpha^2 is irrational, or alpha^2 = p/q with p, q natural numbers and 47 not dividing q, or alpha^2 >= 2, or alpha^2 <= 1/(7 * 47^4 * 48). The closing remarks (p. 12) say the authors believe both bounds are far from optimal and the conditions on alpha can be dropped. The theorems forbid a red ell_3 but not a red unit pair (in the coloring of Theorem 1 every point at distance less than 1 from the origin is red), so they give no bound for problem 188, whose red class must avoid unit pairs.

Source: https://arxiv.org/abs/2402.12567.

Bears on.

• Problem 188: the results are line-versus-line colorings of E^n with a red ell_3 forbidden in place of the problem's red unit pair; they give no bound on the problem's k (see each result page).

Result pages.

• Theorem 1 (p. 2): E^n does not arrow (ell_3, ell_1177), by an explicit spherical coloring modulo 29.
• Theorem 2 (p. 2): E^n does not arrow (ell_3, alpha ell_8649) for alpha in four stated ranges.

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