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This Members' Colloquium talk explores the Erdős unit distance problem, which investigates how many pairs of points in a set of n distinct points in a plane can be at distance 1 from each other. Learn about the history of this challenging problem, from Erdős's 1946 work showing that U(P)=Θ(n1+c/log log n) to the best upper bound of O(n4/3) established in 1984 with no progress since. Discover a completely new approach that connects the unit distance problem to graph rigidity theory, including a novel structure theorem for point configurations with many unit distances and a new conjecture about rigid subgraphs. See how solving this rigidity conjecture could lead to the first improvement in the upper bound for the unit distance problem in four decades. The presentation also covers how to prove a weaker version of the rigidity conjecture by reducing it to a line-line incidence question in ℝ3. Based on joint work with J. Pach and J. Solymosi.
