Problem #213: Let $n\geq 4$. Are there $n$ points in $\mathbb{R}^2$, no...

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Progress

Anning and Erdős [AnEr45] proved there cannot exist an infinite such set. Harborth constructed such a set when $n=5$. The best construction to date, due to Kreisel and Kurz [KK08], has $n=7$.

Ascher, Braune, and Turchet [ABT20] have shown that there is a uniform upper bound on the size of such a set, conditional on the Bombieri-Lang conjecture. Greenfeld, Iliopoulou, and Peluse [GIP24] have shown (unconditionally) that any such set must be very sparse, in that if $S\subseteq [-N,N]^2$ has no three on a line and no four on a circle, and all pairwise distances integers, then\[\lvert S\rvert \ll (\log N)^{O(1)}.\]See also [130].