Arman 2018 result asymmetric euclidean ramsey theory

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theorem_2_1: Proves that every red-blue coloring of three-dimensional space with no two red points at distance one contains five unit-spaced blue collinear points.

theorem_3_1: Proves that every red-blue coloring of three-dimensional space with no two red points at distance one contains six unit-spaced blue collinear points.

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Andrii Arman, Sergei Tsaturian, A result in asymmetric Euclidean Ramsey theory. Discrete Mathematics 341 (2018), no. 5, 1502-1508. DOI 10.1016/j.disc.2017.10.015. arXiv:1702.04799. The copy read for this card is arXiv v1 (15 Feb 2017); the labels and page numbers below are that edition's.

Here l_i is i collinear points with consecutive distance one, and E^n -> (F_1, F_2) says every red-blue coloring of E^n has a red copy of F_1 or a blue copy of F_2. Theorem 2.1 (p. 2) gives a short proof of E^3 -> (l_2, l_5), which Erdos et al. asked about; the paper reports that it also follows from the five-point result E^3 -> (l_2, T_5) proved in Ivan's master's thesis, a result it says was never published. Theorem 3.1 (p. 4) strengthens it to E^3 -> (l_2, l_6). The proofs are geometric propagation arguments. Assuming no red l_2 and no blue l_5, Lemmas 2.2 and 2.3 (pp. 2-3) exclude red pairs at distance 2 and sqrt(7): blue circles around the two red points force a whole circle of red points of radius greater than 1/2, which contains a red unit pair. Section 3 (pp. 4-10), assuming no red l_2 and no blue l_6, excludes an all-blue disk of radius sqrt(3) (Lemma 3.2), red pairs at distances 2, 4 and 3 (Lemmas 3.3-3.5), and, in a unit triangular lattice with two red nodes at distance sqrt(3), any blue l_5 (Lemma 3.6). Both theorems are statements about E^3 only. The introduction (p. 2) recalls Juhasz's planar theorem E^2 -> (l_2, T_4) for every four-point configuration T_4.

Source: https://arxiv.org/abs/1702.04799. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1702.04799), every other right reserved.

Results.

• Theorem 2.1 (p. 2): E^3 -> (l_2, l_5), with Lemmas 2.2 and 2.3 recorded in its proof pointer.
• Theorem 3.1 (p. 4): E^3 -> (l_2, l_6), with Lemmas 3.2-3.6 stated on its page.

Read status. Claims checked: both theorems and the lemmas recorded on their pages were read clause by clause on the arXiv v1 PDF; the proofs were read for structure only.

Bears on.

• #188: the problem is about colorings of the plane. Theorems 2.1 and 3.1 are the three-dimensional arrows E^3 -> (l_2, l_5) and E^3 -> (l_2, l_6). A planar arrow implies the same arrow in E^3, not conversely, so neither theorem gives a bound for the problem.
• #214: the paper's introduction (p. 2) recalls Juhasz's theorem E^2 -> (l_2, T_4) for every four-point configuration T_4, which includes the unit square the problem asks about. The paper's own theorems concern collinear configurations in E^3 and do not address the square.

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