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Source. D. Burt, E. Goldstein, S. Manski, S. J. Miller, E. A. Palsson and H. Suh, Crescent configurations, arXiv:1509.07220v1 [math.CO] (24 September 2015); Theorem 1.3 on p. 2, its proof in Section 2 (p. 3), and the function $\mathcal D(n)$ of Section 3 (pp. 3--4). The copy read is identified on the source card.

Read depth. Claims checked: the statement was read clause by clause on the page images. The proof was read for structure only and not checked; as printed it tracks the distance multiplicities and does not spell out the general-position condition for the added points. Nothing here is independently reviewed.

Statement

Theorem 1.3 (p. 2). "For all $n\ge3$, there exists a set of $n$ points in a crescent configuration in $\mathbb R^{n-2}$."

Crescent configuration is Definition 1.2: general position in $\mathbb R^d$ in the sense of Definition 1.1, with $n-1$ distinct distances occurring exactly $1,2,\ldots,n-1$ times. The paper draws the consequence (p. 2) that for each $n$ some dimension $d$ admits $n$ points in crescent configuration in $\mathbb R^d$.

Section 3 (p. 3) defines $\mathcal D(n)$ as the least dimension greater than $1$ in which $n$ points can be placed in crescent configuration, and notes that the construction gives $\mathcal D(n)\le n-2$ for all $n>3$.

Proof pointer

Section 2, p. 3. By induction, $n-1$ points are placed in crescent configuration in $\mathbb R^{n-2}$, starting from an isosceles triangle that is not equilateral in $\mathbb R^2$. In the inductive step the $n-2$ earlier points lie on a sphere in a hyperplane of $\mathbb R^{n-2}$; the new point is put on the line through the sphere's center perpendicular to that hyperplane, at a distance not yet occurring, so it is equidistant from all earlier points and adds one new distance with multiplicity $n-2$. The $n$th point is the center of the hypersphere through the first $n-1$ points, adding one distance with multiplicity $n-1$; the position of the $(n-1)$st point on its line controls the radius, so the radius can be kept new. The induction starts at $n=4$; the case $n=3$ is the line, where the introduction (p. 1) notes that an arithmetic progression works.

Dependencies

Definitions 1.1 and 1.2 of the same paper.

Bears on

• Problem 217: the problem asks for planar crescent configurations; the theorem constructs them in $\mathbb R^{n-2}$, which is the plane only for $n=4$, and proves nothing about the planar question. Section 3 lists as open whether $\mathcal D(n)$ is bounded (a question it credits to Albujer), sublinear or monotonically increasing, and whether planar constructions exist for $n\ge9$.