Problem #754: Let $f(n)$ be maximal such that there exists a set $A$ of...

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Progress

Avis, Erdős, and Pach [AEP88] proved that\[\frac{n}{2}+2 \leq f(n) \leq (1+o(1))\frac{n}{2}.\]This was proved by Swanepoel [Sw13], who in fact proved more generally that, in any finite set $A\subset \mathbb{R}^4$ of size $n$ and any choice of distance $d(x)$ for each $x\in A$,\[\sum_{x\in A}\sum_{y\in A}1_{\lvert x-y\rvert =d(x)}\leq \tfrac{1}{2}n^2+O(n),\]and proved similar results for finite point sets in higher dimensional space.