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Source. Conjecture 1, p. 4, Section 3, of Andrii Arman and Sergei Tsaturian, Equally spaced collinear points in Euclidean Ramsey theory, arXiv preprint (2017), arXiv:1705.04640, read in arXiv:1705.04640v2 (15 May 2017) as named on the source card; labels and pages here are that version's pp. 1--4.

Statement

The arrow notation and $\ell_i$ are as on the Theorem 2.1 page; $\mathbb E^n\nrightarrow(F_1,F_2)$ means some red/blue colouring of $\mathbb E^n$ has no red copy of $F_1$ and no blue copy of $F_2$.

Conjecture 1 (p. 4). "There is an integer $k$, such that for every integer $n$ $\mathbb{E}^n\nrightarrow(\ell_3,\ell_k)$."

Context (pp. 3--4). The paper notes that the result of Conlon and Fox, and its own Theorem 2.1, imply that for every $k$ some $n$ has $\mathbb E^n\to(\ell_2,\ell_k)$, and that a result of Erdős et al. implies $\mathbb E^n\nrightarrow(\ell_6,\ell_6)$ for all $n$. It asks for the least $s$ such that some $k$ has $\mathbb E^n\nrightarrow(\ell_s,\ell_k)$ for all $n$, and conjectures $s=3$, stated as Conjecture 1. It adds that Conlon and Fox made a similar conjecture.

Later work. Conlon and Wu's Theorem 1.1 proves the statement of Conjecture 1, that one natural number $m$ has $\mathbb E^n\nrightarrow(\ell_3,\ell_m)$ for all $n$.

Read depth. Claims checked: Conjecture 1 and its motivating paragraph were read clause by clause on the page images. Nothing here is independently reviewed.

Bears on

• Problem 188: context only. The conjecture forbids a red $\ell_3$ rather than a red pair at distance one, so colourings of the kind it asks for may contain red unit pairs and give no bound on the least $k$ of the problem. The paper does not connect the conjecture to the problem.