Two New Approaches to Obtaining Estimates in the Danzer–Grünbaum Problem

We use probabilistic methods to estimate the cardinality of a set $S$ in a Euclidean space such that no three points of $S$ form a right or an obtuse angle. Let $a(n)$ be the cardinality of a maximal subset $S\subset\mathbb R^n$ with this property. We prove that
$$ a(n)\ge\frac23\biggl\lfloor\sqrt2\biggl(\frac2{\sqrt3}\biggr)^n\biggr\rfloor. $$