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.
Categories: math.CO math.MG