Finite Point Configurations in the Plane, Rigidity and Erdős Problems

Abstract

For a finite point set E ⊂ ℝd and a connected graph G on k + 1 vertices, we define a G-framework to be a collection of k + 1 points in E such that the distance between a pair of points is specified if the corresponding vertices of G are connected by an edge. We consider two frameworks the same if the specified edge-distances are the same. We find tight bounds on such distinct-distance drawings for rigid graphs in the plane, deploying the celebrated result of Guth and Katz. We introduce a congruence relation on a wider set of graphs, which behaves nicely in both the real-discrete and continuous settings. We provide a sharp bound on the number of such congruence classes. We then make a conjecture that the tight bound on rigid graphs should apply to all graphs. This appears to be a hard problem even in the case of the nonrigid 2-chain. However, we provide evidence to support the conjecture by demonstrating that if the Erd˝os pinned-distance conjecture holds in dimension d, then the result for all graphs in dimension d follows.