On Perles' configuration

Abstract

In the 60s, Micha Perles constructed a point-line arrangement in the plane on nine points, which can not be realized only by points with rational coordinates. Gr\"unbaum conjectured that Perles' construction is the smallest: any geometric arrangement on eight or fewer points if it is realizable with real coordinates in the plane, it is also realizable with rational coordinates. In this paper, we prove the conjecture.