PROVED
This has been solved in the affirmative.
Let $A\subset \mathbb{R}^2$ be a set of size $n$ and let $\{d_1<\ldots<d_k\}$ be the set of distinct distances determined by $A$. Let $f(d)$ be the number of times the distance $d$ is determined. Is it true that\[f(d_1)f(d_k) \leq (\tfrac{9}{8}+o(1))n^2?\]
A question of Erdős and Pach
[ErPa90], who write that an 'easy' unpublished construction of Makai shows that this would be the best possible, and that it is 'not difficult' to prove\[f(d_1)+f(d_k) \leq 3n -c\sqrt{n}+o(\sqrt{n})\]for some $c>0$, and ask what the best possible value of $c$ is.
A stronger version of this problem (which implies the inequality in the problem statement) is that\[f(d_1)\leq 3n-2m+o(\sqrt{n}),\]where $m$ is the number of vertices of the convex hull of $A$.
The odd regular polygon shows that it is possible for $f(d_i)\geq n$ for all $i$.
The original problem was solved by Dumitrescu
[Du19], who proved that\[f(d_1)f(d_k)\leq \frac{9}{8}n^2+O(n).\]See also
[132] and
[756].
View the LaTeX source
Additional thanks to: Adrian Dumitrescu
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #957, https://www.erdosproblems.com/957, accessed 2026-01-16
