Currier 2026 avoiding short progressions euclidean ramsey theory

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proposition_2_4: Currier, Moore and Yip's criterion reducing the statement that every real quadratic k^2 + bk + c, scaled by d and floored, meets a residue set S modulo p for some 0 <= k <= N to finitely many rational cases.

theorem_1_1: Currier, Moore and Yip's red-blue spherical colorings of Euclidean space with no red l_3 and no blue l_20, no red l_4 and no blue l_14, and no red l_5 and no blue l_8, where l_m is m collinear points at unit spacing.

theorem_1_2: Currier, Moore and Yip's three-colorings of Euclidean space avoiding the triples (l_3, l_3, l_8), (l_3, l_4, l_7) and (l_3, l_5, l_5) of unit-spaced collinear configurations, one forbidden in each color.

theorem_1_3: Currier, Moore and Yip's answer to Führer and Tóth: for every positive real alpha, Euclidean space has a red-blue coloring with no red l_3 and no blue 6889-term collinear progression of spacing alpha.

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Gabriel Currier, Kenneth Moore, Chi Hoi Yip, Avoiding short progressions in Euclidean Ramsey theory. Journal of Combinatorial Theory, Series A 217 (2026), 106080, doi:10.1016/j.jcta.2025.106080. arXiv:2404.19233. The copy read for this card is arXiv:2404.19233v3, submitted 30 May 2025.

The paper gives a general framework for constructing red-blue colorings of E^n with no red congruent copy of ell_r and no blue copy of ell_s, where ell_m is m collinear points at consecutive distance 1. Theorem 1.1 proves that for every positive n, E^n does not arrow (ell_3, ell_20), (ell_4, ell_14), or (ell_5, ell_8), improving Fuhrer and Toth's (ell_3, ell_1177) and matching in spirit Erdos et al.'s (ell_6, ell_6); Theorem 1.2 gives three-coloring analogs, that E^n does not arrow (ell_3, ell_3, ell_8), (ell_3, ell_4, ell_7), or (ell_3, ell_5, ell_5). Theorem 1.3 answers a question of Fuhrer and Toth in the positive by proving E^n does not arrow (ell_3, alpha ell_6889) for every real alpha > 0, giving a uniform bound independent of the scaling. The method uses spherical colorings (color depending only on distance from the origin) together with a law-of-cosines lemma and a finite floor-quadratic verification (Corollary 2.2, Proposition 2.4) certifying the absence of short monochromatic progressions. The paper's colorings forbid red ell_3, ell_4 or ell_5, never a red unit pair ell_2, and it notes (p. 2) that spherical colorings always contain monochromatic copies of ell_2; its results therefore give no bound for problem 188, whose red class must avoid unit pairs. Its introduction (p. 1) restates Juhasz's theorem that every two-coloring of the plane with no red unit pair has a blue copy of each four-point configuration, which settles problem 214, and Csizmadia and Toth's eight-point configuration for which this fails; the paper proves nothing new for problem 214.

Source: https://arxiv.org/abs/2404.19233. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2404.19233), every other right reserved.

Bears on.

• Problem 188: the results are line-versus-line colorings of E^n with a red ell_3, ell_4 or ell_5 forbidden in place of the problem's red unit pair; they give no bound on the problem's k (see each result page).
• Problem 214: the introduction (p. 1) cites Juhasz's four-point theorem, which answers the problem, and Csizmadia and Toth's eight-point configuration K' with E^2 not arrowing (ell_2, K'); the paper proves nothing about the problem.

Result pages.

• Theorem 1.1 (p. 2): E^n does not arrow (ell_3, ell_20), (ell_4, ell_14) and (ell_5, ell_8).
• Theorem 1.2 (p. 2): E^n does not arrow (ell_3, ell_3, ell_8), (ell_3, ell_4, ell_7) and (ell_3, ell_5, ell_5).
• Theorem 1.3 (p. 2): E^n does not arrow (ell_3, alpha ell_6889) for every real alpha > 0.
• Proposition 2.4 (p. 3), with Lemma 2.1 and Corollaries 2.2 and 2.3: the finite test that certifies the colorings of Theorems 1.1 and 1.2.

Section 3.3 (pp. 7-8) also gives, without a result page here, red-blue colorings with no red member of the parallelogram family P_gamma (diagonals alpha and beta with (alpha^2 - beta^2)/2 = gamma) and no blue unit progression: (P_1, ell_18), (P_2, ell_20), (P_3, ell_19) and (P_4, ell_21), where ell_3 is in P_2 and ell_4 is in P_4.

Read status. Claims checked for the four result pages: statements, hypotheses, labels and pages were read on the printed arXiv:2404.19233v3 pages. Proofs were read but not checked step by step, and the computer checks the paper relies on were not rerun.

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