Growth Rate of the Number of Empty Triangles in the Plane

Given a set P of n points in the plane, in general position, denote by N_Δ(P) the number of empty triangles with vertices in P. In this paper we investigate by how much N_Δ(P) changes if a point x is removed from P. By constructing a graph GP(x) based on the arrangement of the empty triangles incident on x, we transform this geometric problem to the problem of counting triangles in the graph GP(x). We study properties of the graph GP(x) and, in particular, show that it is kite-free. This relates the growth rate of the number of empty triangles to the famous Ruzsa-Szemer\'edi problem.