Asked by Erdős and Pach in the stronger form of whether all distances (aside from the largest) can occur with multiplicity $>n$. Hopf and Pannowitz
[HoPa34] proved that the largest distance between points of $A$ can occur at most $n$ times, but it is unknown whether a second such distance must occur (see
[132]).
The answer is yes: Bhowmick
[Bh24] constructs a set of $n$ points in $\mathbb{R}^2$ such that $\lfloor\frac{n}{4}\rfloor$ distances occur at least $n+1$ times. More generally, they construct, for any $m$ and large $n$, a set of $n$ points such that $\lfloor \frac{n}{2(m+1)}\rfloor$ distances occur at least $n+m$ times.
Further investigations were undertaken by Clemen, Dumitrescu, and Liu
[CDL25], who proved, amongst other results, that if we take $A$ to be the square grid of $n$ points, then there are at least $n^{c/\log\log n}$ many distances which occur with multiplicity at least $n^{1+c/\log\log n}$ (for some constant $c>0$).
See also
[132] and
[957].
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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 #756, https://www.erdosproblems.com/756, accessed 2026-04-11
