Problem 956

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claims/: The 3 claim pages of Problem 956, one per claimant's result; the problem's standing derives from them.

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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 $n$.

Status. Open. The site labels the problem OPEN; two pending claims of the order $n^{4/3}$, Valtr's announcement of 2005 and Chojecki's note of 2026, are recorded on their claim pages, listed in the Current assessment.

Source. erdosproblems.com/956, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #956, https://www.erdosproblems.com/956.

References.

• [ErPa90] Erdős, P. and Pach, J., Variations on the theme of repeated distances. Combinatorica (1990), 261-269.

Formalization. The formal-conjectures catalog has no statement file for the problem. Two Lean developments of lower bounds are linked from Chojecki's claim page; neither has been built here.

Current assessment

Claimed. Erdős and Pach proved $h(n)=O(n^{4/3})$, recorded on the accepted partial claim page. For the lower bound, a single point is a compact convex set, so $h(n)$ is at least the largest number of unit distances among $n$ planar points, as the site remarks; with the accepted claim on Problem 90 this gives $h(n)\geq n^{1+\delta}$ for a fixed $\delta>0$ and infinitely many $n$ (a remark of this page), which is short of all large $n$. Valtr announced $h(n)=\Theta(n^{4/3})$ in an Oberwolfach abstract of 2005 without a published proof, recorded on its claim page. Chojecki's note of 27 April 2026, whose result he attributed to GPT-5.5 Pro, gives an explicit parabolic-cap construction with $h(n)\geq c_0n^{4/3}$ and hence $h(n)=\Theta(n^{4/3})$, recorded on its claim page together with Aristotle's partial Lean file and Linmiao Xu's Lean development of lower bounds, registered in the Palomar registry on 4 October 2026. Neither claim is refereed or independently reviewed, so the standing is a pending full claim. Search scope, 2026-10-07: the site's problem page and discussion thread, the Oberwolfach report, the claimant's note and Lean file, the Lean repository and registry record, and the formal-conjectures catalog; no forum proof claim, release item or lead names the problem.