Arman 2017 equally spaced collinear points euclidean ramsey

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conjecture_1: Arman and Tsaturian's conjecture that some integer k gives, in every dimension n, a red/blue colouring of E^n with no red l_3 and no blue l_k.

theorem_2_1: Arman and Tsaturian's theorem that for every integer k at least four, each red/blue colouring of E^k has two red points at distance one or k+3 blue collinear points with consecutive distance one.

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Andrii Arman, Sergei Tsaturian, Equally spaced collinear points in Euclidean Ramsey theory. arXiv preprint (2017). arXiv:1705.04640. The copy read for this card is arXiv v2 (15 May 2017).

Theorem 2.1 (p. 2) proves E^k -> (l_2, l_{k+3}) for every integer k >= 4, where l_i is i collinear points at consecutive distance one; equivalently m(k) >= k+3 for the largest m with E^k -> (l_2, l_m). The abstract calls the result new for 4 <= k <= 10, and the introduction (p. 1), after recalling the Conlon-Fox estimate (1+o(1))1.2^k < m(k) < 10^{5k}, calls it a better bound for small values of k, namely k <= 10. The method is spherical: Lemma 2.2 (p. 2) finds a blue unit simplex Delta^{k-2} on any (k-2)-dimensional sphere of radius sqrt(3)/2 in E^{k-1} coloured with no red unit pair, by a hypercap propagation argument, and Lemma 2.3 (p. 3) shows that in a colouring of E^k with no red l_2, two red points at an integer distance d with 2 <= d <= k+1 force a blue l_{k+3}. The authors state (p. 1) that the techniques do not apply when k <= 3, so the note does not imply E^2 -> (l_2, l_5) or E^3 -> (l_2, l_6). Conjecture 1 (p. 4), in the concluding remarks, conjectures an integer k with E^n -/-> (l_3, l_k) for every n, motivated by the fact, which the paper says a result of Erdos et al. implies, that E^n -/-> (l_6, l_6) for all n; Conlon and Wu later proved this statement (their Theorem 1.1).

Version used: the labels and pages here are those of arXiv:1705.04640v2, pp. 1--4. Read status: claims checked for Theorem 2.1, Lemmas 2.2 and 2.3 and Conjecture 1, read clause by clause on the page images; the proofs (pp. 2--3) were read through, not verified. Nothing here is independently reviewed.

Source: https://arxiv.org/abs/1705.04640. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1705.04640), every other right reserved.

Bears on. #188: context only. Theorem 2.1 (p. 2) gives E^k -> (l_2, l_{k+3}) only for k >= 4, and the paper states (p. 1) that it does not imply E^2 -> (l_2, l_5), so it gives no bound on the least k of the problem, which asks about the plane. The introduction (p. 1) recalls Erdos and Graham's claim that m(2) exists, the question of Erdos et al. whether E^2 -> (l_2, l_5), and Tsaturian's proof that E^2 -> (l_2, l_5); that recalled result is Tsaturian's, not this paper's. Conjecture 1 (p. 4) forbids a red l_3 rather than a red unit pair and so gives no bound on the problem's k.

Contents.

• Theorem 2.1 (p. 2): For every integer k >= 4, E^k -> (l_2, l_{k+3}): any red-blue coloring of k-space has two red points at distance one or k+3 equally spaced blue collinear points.
• Lemma 2.2 (p. 2): For k >= 4, if E^{k-1} is coloured with no two red points at distance one, then every (k-2)-dimensional sphere of radius sqrt(3)/2 contains a blue unit simplex Delta^{k-2}.
• Lemma 2.3 (p. 3): If E^k has no red l_2 and contains two red points at integer distance d with 2 <= d <= k+1, then it contains a blue l_{k+3}.
• Conjecture 1 (p. 4): There is an integer k such that E^n -/-> (l_3, l_k) for every integer n; the paper presents it as the conjecture that the least s admitting such a k for all n is 3.

Results.

• Theorem 2.1 (p. 2): E^k arrows (l_2, l_{k+3}) for every k >= 4; the page also states Lemmas 2.2 and 2.3.
• Conjecture 1 (p. 4): one k with E^n not arrowing (l_3, l_k) for every n.

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