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.