Non-Euclidean Erdős–Anning theorems.
D. Eppstein.
arXiv:2401.06328.
41st International Symposium on Computational Geometry (SoCG 2025), Kanazawa, Japan.
Leibniz
International Proceedings in Informatics (LIPIcs) 332, 2025,
pp. 46:1–46:15, doi:10.4230/LIPIcs.SoCG.2025.46.
J. Computational Geometry 17 (2): 46–76, 2026, doi:10.20382/jocg.v17i2a2 (special issue for SoCG 2025).
Non-collinear point sets with integer distances must be finite, for strictly convex distance functions on the plane, for two-dimensional complete Riemannian manifolds of bounded genus, and for geodesic distance on convex surfaces in 3d.
For related results on higher-dimensional Euclidean and hyperbolic spaces, see "Tangent spheres and integer distances".
(SoCG'25 talk slides – Blog post: Integer distances in floppy metric spaces)