Burt 2015 crescent configurations

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definition_1_2: The paper's name for n points in general position in R^d (Definition 1.1) that determine n-1 distinct distances, the i-th occurring exactly i times for each i from 1 to n-1; for d = 2 it is the condition of Problem 217.

remark_3_1: The authors' report of an exhaustive computer search of a 91-point hexagonal region of the triangular lattice that found no crescent configuration of nine points, an instance of Problem 217 left open.

theorem_1_3: For every n at least 3 there are n points in general position in (n-2)-dimensional space whose n-1 distinct distances occur exactly 1, 2, ..., n-1 times; a higher-dimensional analogue of Problem 217 that says nothing about the plane.

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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, 4 pp.; MSC 52C10, 52C35. A journal version appeared in Integers 16 (2016), #A38 (DOI 10.5281/zenodo.10606286); it was not read or compared with the arXiv copy.

The copy read for this card is the arXiv build of version 1 (the margin stamp reads "arXiv:1509.07220v1 [math.CO] 24 Sep 2015"; the date footnote on p. 1 reads November 5, 2018, evidently the date this PDF was built), 4 pages with a text layer. Provenance: obtained in September 2026 through the survey download; the download URL was not recorded, but the stamp identifies the copy as https://arxiv.org/abs/1509.07220v1. 339,730 bytes. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1509.07220), every other right reserved.

Read status: claims checked. Definitions 1.1 and 1.2, Theorem 1.3 and Remark 3.1 were read clause by clause on the page images; the proof of Theorem 1.3 (section 2, p. 3) was read for structure but not checked.

Result pages: Definition 1.2 (with Definition 1.1), Theorem 1.3 and Remark 3.1.

Contents

• Definition 1.1 (p. 2): $n$ points are in general position in $\mathbb R^d$ if no $d+1$ lie on a hyperplane and no $d+2$ on a hypersphere. Definition 1.2 (p. 2): $n$ points are in crescent configuration in $\mathbb R^d$ if they are in general position and determine $n-1$ distinct distances such that for every $1\le i\le n-1$ some distance occurs exactly $i$ times. For $d=2$ this is the condition of #217.
• Page 2 recalls that Erdős conjectured in 1989 (the paper's [Erd89]) that no planar crescent configuration exists for large $n$, that Palásti ([Pal87], [Pal89]) gave the constructions for $n=7$ and $n=8$ on the triangular lattice, and that no construction for $n\ge9$ is known. Figure 1 (p. 2) gives coordinates for Palásti's eight-point example.
• Theorem 1.3 (p. 2; proof in section 2, p. 3): for all $n\ge3$ there is a crescent configuration of $n$ points in $\mathbb R^{n-2}$. The proof builds $n-1$ points by induction, adding at each step a point on the line through the center of the sphere of the previous points, perpendicular to their hyperplane, at a new distance, so that the new point is equidistant from all earlier points; the $n$th point is the center of the hypersphere through the first $n-1$, with the radius chosen to avoid earlier distances.
• Section 3 (pp. 3--4): with $D(n)$ the least dimension greater than 1 in which $n$ points can form a crescent configuration, the construction gives $D(n)\le n-2$ for $n>3$; the paper lists as open whether $D(n)$ is bounded (Albujer's question, [Alb]), sublinear or monotone, whether $D(n)=k$ implies crescent configurations of $n$ points in $\mathbb R^d$ for every $d>k$ (embedding need not keep general position), and whether planar constructions for $n\ge9$ exist (asked before in [CFG] and [Alb]), on the triangular lattice or otherwise. Remark 3.1 (p. 4): an exhaustive search of a 91-point hexagonal region of the triangular lattice found no crescent configuration for $n=9$ (over 900 hours of computation).

Compiled scope

Only the statements above were checked against the page images. The proof of Theorem 1.3 was read for structure, 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.

Bears on. #217, as the paper the problem page names in prose: it constructs crescent configurations of every size $n\ge3$ in $\mathbb R^{n-2}$ (Theorem 1.3), leaving open whether $D(n)$, the least dimension greater than 1 in which $n$ points can form a crescent configuration, is bounded, and records a search of a 91-point region of the triangular lattice that found no nine-point planar example (Remark 3.1); it proves nothing about the planar question. Its Definition 1.2 for $d=2$ is the problem's condition.

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