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Claim. There are six points in the plane, no three on a line, no four on a circle and no point equidistant from three others, that determine five distinct distances occurring once, twice, three, four and five times, and that contain no equilateral triangle. This is the result of I. Palásti, A distance problem of P. Erdős with some further restrictions, Discrete Math. 76 (1989), no. 2, 155–156, as the zbMATH review (Zbl 0669.52007) states it. The further restrictions answer a question Erdős asked, by the paper's account, when he saw the seven-point example of [[problems/distance_problems/E0217/claims/1987_01_01_palasti|Palásti's 1987 paper]]: whether such a set of five points exists with no equilateral triangle; the paper gives six points with no equilateral triangle. In the notation of Problem 217, the answer is yes for $n=6$. Erdős's 1983 lecture reports an earlier unpublished six-point example by a Hungarian high-school student (card).

Covers. The instance $n=6$, with the further restrictions stated. The property of the problem does not pass to subsets, so nothing is claimed for any other $n$; the instances $n=5$, $7$ and $8$ are settled on Pomerance's, [[problems/distance_problems/E0217/claims/1987_01_01_palasti|Palásti's seven-point]] and eight-point claim pages.

Depends on. Nothing in this wiki.

Acceptance. Refereed: Discrete Mathematics 76 (1989), no. 2, 155–156; the Crossref record of the DOI gives these data. The site's remarks credit the six-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.