Every large point set has an obtuse angle

Abstract

Around 1950 Paul Erdős conjectured that every set of more than 2dpoints in ℝddetermines 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 ℝdwhich only has acute angles (including right angles) has size at most 2d. 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.