Exponentially sized pointsets with angles less than 61 degrees

TL;DR

This paper establishes exponential bounds on the size of point sets in high-dimensional space where all angles are less than a specified threshold slightly above 60 degrees, using geometric and combinatorial methods.

Contribution

It provides tight exponential bounds for the maximum size of such point sets, refining previous approaches and connecting geometric packing problems with hypergraph intersection problems.

Findings

Point set size grows as (1+Theta(c))^d for small c

Upper bounds relate to sphere cap packing problems

Lower bounds involve hypergraph intersection problems

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 Erd\H{o}s and F\"{u}redi. 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.