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Claim. Among the lattice points of the tiling of the plane by equilateral triangles there are, for every $n$ from $3$ to $8$, $n$ points, no three on a line and no four on a circle, that determine $n-1$ distinct distances with the $i$-th occurring $i$ times. This is the result of I. Palásti, Lattice-point examples for a question of Erdős, Period. Math. Hungar. 20 (1989), no. 3, 231–235, as the zbMATH review (Zbl 0687.52004) states it: examples were known for $n=3,\ldots,7$, and the paper finds examples for $n=3,\ldots,8$ on the triangular lattice, so the eight-point example is new. Burt, Goldstein, Manski, Miller, Palsson and Suh give its coordinates in Figure 1 of Crescent configurations (card). In the notation of Problem 217, the answer is yes for $n=8$.

Covers. The instance $n=8$, with lattice examples for every $n\le7$. The property of the problem does not pass to subsets, so each example settles only its own $n$; the instances $n=5$, $6$ and $7$ are also settled on Pomerance's, [[problems/distance_problems/E0217/claims/1989_01_01_palasti|Palásti's six-point]] and seven-point claim pages. Not covered: every $n\ge9$, for which no example is known.

Depends on. Nothing in this wiki.

Acceptance. Refereed: Periodica Mathematica Hungarica 20 (1989), no. 3, 231–235, September 1989; the Crossref record of the DOI gives these data. The site's remarks credit the eight-point construction to this paper, but the site labels the problem OPEN, so that credit is commentary on an open problem and no reviewed evidence is listed.