Exponentially sized pointsets with angles less than 61 degrees

Abstract

We prove that any set of points in R d , any three of which form an angle less than π 3 + c, has size (1 + Θ(c)) d for sufficiently small c > 0. The proof is based on a refinement of an approach by Erdős and Füredi. The lower bound is relying on a problem about large hypegraphs with small edge intersections, while the upper bound is tightly connected to the problem of packing disjoint caps on a sphere.