Harborth's theorem p(5)=18 revisited

Abstract

Let p(n) be the minimum cardinality of a finite planar point set in which each point has exactly n unit-distance neighbours. Harborth proved p(5)=18 in work presented in 1985 and published in 1987, using a detailed case analysis. We revisit the proof using convex-hull arguments, incidence counts, and elementary properties of intersections of unit circles. These tools give an alternative treatment of configurations with $14$ and $16$ points and organize the final case distinction by a common vector-sum structure. We also treat a configuration not explicitly covered by the case analysis in the online PDF copy of Harborth's paper available to us, and give an explicit justification of the six-point classification used in the argument. No result subsequent to Harborth's proof is required.

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BibTeX RIS

BibTeX

@misc{winkler2026harborths,
  title = {{Harborth's theorem p(5)=18 revisited}},
  author = {Mike Winkler},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40725},
  eprint = {2610.40725},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40725},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Harborth's theorem p(5)=18 revisited
AU  - Mike Winkler
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40725
UR  - https://arxiv.org/abs/2610.40725
AB  - Let p(n) be the minimum cardinality of a finite planar point set in which each point has exactly n unit-distance neighbours. Harborth proved p(5)=18 in work presented in 1985 and published in 1987, using a detailed case analysis. We revisit the proof using convex-hull arguments, incidence counts, and elementary properties of intersections of unit circles. These tools give an alternative treatment of configurations with $14$ and $16$ points and organize the final case distinction by a common vector-sum structure. We also treat a configuration not explicitly covered by the case analysis in the online PDF copy of Harborth's paper available to us, and give an explicit justification of the six-point classification used in the argument. No result subsequent to Harborth's proof is required.
ER  -

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