Every large point set has an obtuse angle

Abstract

Around 1950 Paul Erdős conjectured that every set of more than 2^d points in ℝ^d determines at least one obtuse angle, that is, an angle that is strictly greater than $\frac{\pi}{2}$. In other words, any set of points in ℝ^d which only has acute angles (including right angles) has size at most 2^d. This problemwas posed as a “prize question” by the Dutch Mathematical Society — but solutions were received only for d = 2 and for d = 3.