Finite point configurations in the plane, rigidity and Erdos problems

TL;DR

This paper investigates the number of distinct configurations of finite point sets in the plane based on specified distances, introducing new bounds and conjectures related to rigidity, congruence classes, and Erdős distance problems.

Contribution

It establishes tight bounds on congruence classes of point configurations for rigid graphs and proposes a conjecture extending these bounds to all graphs, supported by connections to Erdős distance conjectures.

Findings

Tight bounds on the number of congruence classes for rigid graphs.

Introduction of a congruence relation with favorable properties.

Evidence linking Erdős pinned-distance conjecture to general graph configurations.

Abstract

For a finite point set $E\subset \mathbb{R}^d$ 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 the 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 non-rigid 2-chain. However we provide evidence to support the conjecture by demonstrating that if the Erd\H os pinned-distance conjecture holds in dimension $d$ then the result for all graphs in dimension $d$ follows.