Erdős Problem #956

The problem

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 $n$.

Statement quoted from erdosproblems.com/956, maintained by Thomas Bloom. Original source: [ErPa90]. Current status there: open.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

∎