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Claim. There are seven points in the plane, no three on a line and no four on a circle, all of whose pairwise distances are integers. Theorem 1 of T. Kreisel and S. Kurz, There Are Integral Heptagons, no Three Points on a Line, no Four on a Circle, Discrete Comput. Geom. 39 (2008), no. 4, 786–790, states more: the minimum diameter of such a seven-point set is $22270$. The proof is an exhaustive, isomorph-free generation of plane integral point sets in general position by increasing diameter, which found a single example at diameter $22270$ and none below it; the paper gives its distance matrix and an exact coordinate embedding. The source card kreisel_2008_there_are_integral_heptagons_no_three summarizes the method. In the notation of Problem 213, the answer is yes for $n=7$.

Covers. The instances $4\le n\le7$, answered yes: any subset of the heptagon keeps all three properties. Not covered: every $n\ge8$, for which no construction is known. The construction supersedes [[problems/distance_problems/E0213/claims/1971_01_01_harborth|Harborth's five points]]. The formal-conjectures statement records the case $n=7$ as the variant erdos_213.variants.KK08, tagged research solved and citing this paper; it carries no proof and is not a formalization.

Depends on. Nothing in this wiki.

Acceptance. Refereed: Discrete & Computational Geometry 39 (2008), no. 4, 786–790, published online 2007-09-29; the Crossref record of the DOI gives these data. The site's remarks credit the largest known construction, seven points, 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.