Two Low-Multiplicity Distances in Planar Point Sets

Definitions and mathematical statement

Let $n\in{\mathbb{N}}$ with $n>4$, and let $P\subset{\mathbb{R}}^2$ be a set of $n$ distinct points, with Euclidean distance $\|p-q\|_2$. Define the set of distances determined by $P$ by \[ D(P)=\{\|p-q\|_2:p,q\in P,\ p\ne q\}. \] For each $d\in D(P)$, define its multiplicity by \[ m_P(d)=\#\left\{\{p,q\}:p,q\in P,\ p\ne q,\ \|p-q\|_2=d\right\}. \] Thus each unordered pair of distinct points is counted once, and every distance in $D(P)$ has positive multiplicity.

Question. Is it true that for every integer $n>4$ and every such set $P$, there exist $d_1,d_2\in D(P)$ such that \[ d_1\ne d_2,\qquad 1\le m_P(d_1)\le n,\qquad 1\le m_P(d_2)\le n? \] The question imposes no convexity or general-position assumption on $P$. The supplied background mentions results for convex or "not-too-convex" sets, but does not define the latter class or state the precise hypotheses of those results; those qualifications cannot be formalized further from this record.

Short English statement

Must every set of $n>4$ distinct points in the Euclidean plane determine two different positive distances, each realized by at most $n$ unordered pairs of points? The supplied record labels the general question open; this status is not independently verified here.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006556$, title: Two Low-Multiplicity Distances in Planar Point Sets; source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2026 but names no proposer.
• Combinatorics (2026), Oberwolfach Reports, Report 1/2026, pp. 5--79, DOI: https://doi.org/10.4171/owr/2026/1. This is the supplied report attribution; no exact page for this question is supplied. The registry labels the statement corrected_verified and reports that the official report was checked, the question isolated from surrounding exposition, and its notation repaired. These are supplied verification claims, not checks performed for this draft.
• Clemen--Dumitrescu--Liu, On the multiplicities of interpoint distances, https://arxiv.org/abs/2505.04283.
• Supplied literature assessment dated exactly 2026-08-22: the distance-multiplicity paper is reported to establish the claim for convex or "not-too-convex" sets, while the universal assertion for arbitrary planar point sets is reported to remain open. The supplied sources checked are the Oberwolfach report and the Clemen--Dumitrescu--Liu paper linked above. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; time-sensitive status and the cited special-case claims require independent verification.