Erdős Problem #956 — 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…

Erdős and Pach asked for the maximum number of unit set-distances among pairwise disjoint translates of one planar compact convex set. The claimed stronger answer is that this maximum grows on the order of four-thirds powers of $n$.

Known results

• Erdős and Pach proved the upper bound $h(n)=O(n^{4/3})$.
• Erdős and Pach proved the related upper bound $O(n^{7/5})$ for arbitrary pairwise disjoint convex sets.
• The elementary comparison $h(n)\ge f(n)$ is recorded, where $f(n)$ is the ordinary point-set unit-distance maximum.

Recent claimed solution

A linked note gives a parabolic-grid construction with $n_k=2k^3+O(k^2)$ and $M_k=\frac{5}{12}k^4+O(k^3)$, claiming $h(n)=\Theta(n^{4/3})$. The discussion attributes the claim to GPT-5.5 Pro and Lean formalization to Aristotle, but notes that the disjointness and Euclidean unit-distance conversion beyond Valtr’s construction still require independent checking.

Current status (as of June 2026): The $O(n^{4/3})$ upper bound is established, while a matching lower bound and hence $h(n)=[0m\Theta(n^{4/3})$ remain an unverified claim.