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 \(N_\varDelta (P)\) the number of empty triangles with vertices in P. In this paper we investigate by how much \(N_\varDelta (P)\) changes if a point x is removed from P. By constructing a graph \(G_P(x)\) 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 \(G_P(x)\) . We study properties of the graph \(G_P(x)\) and, in particular, show that it is kite-free. This relates the growth rate of the number of empty triangles to the famous Ruzsa-Szemerédi problem.