On Some Non-Rigid Unit Distance Patterns

Abstract

AbstractA recent generalization of the Erdős Unit Distance Problem, proposed by Palsson, Senger, and Sheffer, asks for the maximum number of unit distance paths with a given number of vertices in the plane and in 3-space. Studying a variant of this question, we prove sharp bounds on the number of unit distance paths and cycles on the sphere of radius $$1/{\sqrt{2}}$$ 1 / 2 . We also consider a similar problem about 3-regular unit distance graphs in $$\mathbb {R}^3$$ R 3 .