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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