Exponentially sized pointsets with angles less than 61 degrees

Abstract

We prove that any set of points in ${\mathbb{R}}^d$, any three of which form an angle less than $\frac{\pi }{3}+c$, has size $(1+\Theta (c){)}^d$ 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.

arXiv comment6 pages, minor changes from previous version