OPEN
This is open, and cannot be resolved with a finite computation.
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, and suppose the $d_i$ are ordered such that\[f(d_1)\geq f(d_2)\geq \cdots \geq f(d_k).\]Estimate\[\max (f(d_1)-f(d_2)),\]where the maximum is taken over all $A$ of size $n$.
Disclaimer: The open status of this problem reflects the current belief of the owner of this website. There may be literature on this problem that I am unaware of, which may partially or completely solve the stated problem. Please do your own literature search before expending significant effort on solving this problem. If you find any relevant literature not mentioned here, please add this in a comment.
More generally, one can ask about\[\max (f(d_r)-f(d_{r+1})).\]Clemen, Dumitrescu, and Liu
[CDL25], have shown that\[\max (f(d_1)-f(d_2))\gg n\log n.\]More generally, for any $1\leq k\leq \log n$, there exists a set $A$ of $n$ points such that\[f(d_r)-f(d_{r+1})\gg \frac{n\log n}{r}.\]They conjecture that $n\log n$ can be improved to $n^{1+c/\log\log n}$ for some constant $c>0$.
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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 #959, https://www.erdosproblems.com/959, accessed 2026-04-11
