Exceptional Points in Favourite Distance Digraphs

Definitions and mathematical statement

A provisional conventional framework is the following. Let $S\subset{\mathbb{R}}^3$ be a finite set and let $r:S\to(0,\infty)$ assign a favourite distance to each point. The associated directed graph has vertex set $S$ and arc set \[ A(S,r)=\{(x,y)\in S\times S:x\ne y,\ \|x-y\|_2=r(x)\}. \] For a positive integer $n$, one possible meaning of extremality is that $|S|=n$ and $|A(S,r)|$ is maximal over all such pairs with $n$ points. These definitions provide a provisional interpretation; the supplied record does not specify the admissible configurations or the extremal objective explicitly.

The supplied question asks whether the exceptional set $T$ in the extremal favourite-distance digraph problem in ${\mathbb{R}}^3$ is always empty. Its dated literature assessment describes $T$ as having at most two points and says that its emptiness is conjectured.

Unresolved formulation. The record does not define $T$, state the structural result that introduces it, or specify the range of $n$. It also does not say whether emptiness is required for every permitted structural decomposition or whether every extremal configuration merely admits a decomposition with no exceptional points. Consequently, the intended assertion $T=\varnothing$ cannot be given complete hypotheses and quantifiers from the supplied evidence. In particular, no theorem asserting a decomposition or identifying its nonexceptional part is assumed here.

Short English statement

The question asks whether extremal favourite-distance digraphs in ${\mathbb{R}}^3$ always have no exceptional points. The supplied assessment mentions at most two possible exceptional points, but their definition, the relevant structural decomposition, and the applicable configuration sizes are missing. A precise conjecture therefore requires the underlying source formulation.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001892$, title Exceptional Points in Favourite Distance Digraphs, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The record lists proposed year 2011 and names no proposer.
• Supplied report citation: Discrete Geometry (2012), Oberwolfach Reports, DOI 10.4171/owr/2011/44, pp. 2459--2548. https://doi.org/10.4171/owr/2011/44. The background describes these reports as documenting mathematical workshops and their open-problem discussions. The literature-source label is "OWR 44 (2012)."
• Supplied literature source labelled "Swanepoel": https://researchonline.lse.ac.uk/id/eprint/104165/4/Swanepoel_favourite_distances_in_3_space_published.pdf. The supplied assessment refers to a 2020 primary paper stating that the exceptional set of at most two points is conjectured to be empty; the record supplies no theorem number or relevant passage.
• Literature checked at: 2026-08-21. This is supplied dated curation, not independently verified. The supplied status is "open," and the background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified; its verification note says only that TeX notation was enclosed in mathematical delimiters without changing the mathematical content. This does not establish completeness or mathematical correctness of the formulation.