Problem Statement
If $C,D\subseteq \mathbb{R}^2$ then the distance between $C$ and $D$ is defined by\[\delta(C,D)=\inf_{\substack{c\in C\\ d\in D}}\| c-d\|.\]Let $h(n)$ be the maximal number of unit distances between disjoint convex translates. That is, the maximal $m$ such that there is a compact convex set $C\subset \mathbb{R}^2$ and a set $X$ of size $n$ such that all $(C+x)_{x\in X}$ are disjoint and there are $m$ pairs $x_1,x_2\in X$ such that\[\delta(C+x_1,C+x_2)=1.\]Determine $h(n)$ - in particular, prove that there exists a constant $c>0$ such that $h(n)>n^{1+c}$ for all large $n$.
Categories:
Geometry Distances Convex
Progress
A problem of Erdős and Pach [ErPa90], who proved that $h(n) \ll n^{4/3}$. They also consider the related function where we consider $n$ disjoint convex sets (not necessarily translates), for which they give an upper bound of $\ll n^{7/5}$.It is trivial that $h(n)\geq f(n)$, where $f(n)$ is the maximal number of unit distances determined by $n$ points in $\mathbb{R}^2$ (see [90]).
Source: erdosproblems.com/956 | Last verified: January 19, 2026