Tangent spheres and integer distances.
D. Eppstein.
arXiv:2606.18569.
Proc. 38th Canadian Conference on Computational Geometry, 2026,
to appear.
Whenever a system of spheres or hyperspheres in three or more dimensions has the property that it can be partitioned into two subsets with each sphere in one subset externally tangent to each sphere in the other subset, it must be the case that within each subset the spheres all have coplanar centers. We use this to prove a high-dimensional version of the Erdős–Anning theorem: in any Euclidean or hyperbolic space, any point set within which all distances are integers must be either finite or collinear.