On the cardinality of sets in ${\bf R}^d$ obeying a slightly obtuse angle bound

In this paper we explicitly estimate the number of points in a subset $A \subset \R^{d}$ as a function of the maximum angle $\angle A$ that any three of these points form, provided $\angle A < \theta_d := \arccos(-\frac 1 {d}) \in (\pi/2,\pi)$. We also show $\angle A < \theta_d$ ensures that $A$ coincides with the vertex set of a convex polytope. This study is motivated by a question of Paul Erdős and indirectly by a conjecture of László Fejes Tóth.