Erdős originally conjectured this (in [Er46b]) with no 3 vertices equidistant,
but Danzer found a convex polygon on 9 points such that every vertex has three
vertices equidistant from it (but this distance depends on the vertex).
Danzer's construction is explained in [Er87b].
[Er46b] Erdős, P., _On sets of distances of $n$ points_. Amer. Math. Monthly (1946), 248-250.
[Er87b] Erdős, P., _Some combinatorial and metric problems in geometry_. Intuitive geometry (Siófok, 1985), 167-177.
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