Every large point set has an obtuse angle

Abstract

Around 1950 Paul Erdös conjectured that every set of more than 2^dpoints 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^dwhich only has acute angles (including right angles) has size at most 2^d. This problem was posed as a “prize question” by the Dutch Mathematical Society — but solutions were received only for d = 2 and for d = 3.