On sets of points that determine only acute angles (Q1024284)

The authors prove that for every integer \(d\) there is a set of points in the \(d\)-dimensional Euclidean space \({\mathbb E}^d\) of size \(\Omega ((\frac{2}{\sqrt{3}})^d\sqrt{d})\) such that every angle determined by three points in the set is acute, that is, smaller than \(\pi/2\). This result improves on the best known lower bound by a factor of \(\Theta (\sqrt{d})\). The short and elegant proof is probabilistic in nature and it uses a theorem of \textit{C. Bertram-Kretzberg} and \textit{H. Lefmann} [SIAM J. Comput. 29, No. 1, 201--230 (1999; Zbl 0937.68056)] about \(3\)-uniform hypergraphs.