Problem 956

Erdős Problem #956: declared status 'open'. Formalized: no. 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 $n$. Current best: They also consider the related function where we consider $n$ disjoint convex sets (not necessarily translates), for which they give an upper bound of $\ll n^{7/5}$. Prize: no. Tags: convex, distances, geometry.