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Claim. There are five points in the plane, no three on a line and no four on a circle, determining four distinct distances that occur $4$, $3$, $2$ and $1$ times: the instance $n=5$ of Problem 217, answered yes. The construction is described by Erdős in his 1983 lecture transcript Combinatorial problems in geometry, Math. Chronicle 12 (1983), 35–54, p. 54 (the paper link; carded as erdos_1983_combinatorial_problems_geometry), where he credits it to Pomerance and says it corrected his own belief that no example with more than four points exists. Take a unit equilateral triangle $OAB$, its circumcenter $C$, and a point $D$ on the unit circle about $O$ with $CD=BD$. The unit distance occurs four times ($OA$, $OB$, $AB$, $OD$), the circumradius $1/\sqrt3$ three times ($OC$, $AC$, $BC$), the distance $CD=BD$ twice and $AD$ once; the transcript states that no three of the points are collinear and no four concyclic. Either of the two choices of $D$ works.

Covers. The instance $n=5$. The property of the problem does not pass to subsets, so nothing is claimed for any other $n$. The transcript adds that a Hungarian high-school student had found a six-point example, unpublished; the instances $n=6$, $7$ and $8$ are settled on [[problems/distance_problems/E0217/claims/1989_01_01_palasti|Palásti's six-point]], seven-point and eight-point claim pages.

Depends on. Nothing in this wiki.

Standing. Claimed. The construction's only publication is Erdős's lecture transcript, not a refereed paper by the claimant, and the elementary check of its distances is not an outside review; the site's remarks credit the example to Pomerance on a problem the site labels OPEN, which is commentary and not acceptance.