Iterated Point-Line Configurations Grow Doubly-Exponentially

Begin with a set of four points in the real plane in general position. Add to this collection the intersection of all lines through pairs of these points. Iterate. Ismailescu and Radoičić (2003) showed that the limiting set is dense in the plane. We give doubly exponential upper and lower bounds on the number of points at each stage. The proof employs a variant of the Szemerédi-Trotter Theorem and an analysis of the “minimum degree” of the growing configuration. Consider the iterative process of constructing points and lines in the real plane given by the following: begin with a set of points P1 = {p1, p2, p3, p4} in the real plane in general position. For each pair of points, construct the line passing through the pair. This will create a set of lines L1 = {`1, `2, `3, `4, `5, `6}. Some of these constructed lines will intersect at points in the plane that do not belong to the set P1. Add any such point to the set P1 to get a new set P2. Now, note that there exist some pairs of points in P2 that do not lie on a line in L1, namely some elements of P2 \ P1. Add these missing lines to the set L1 to get a new set L2. Iterate in this manner, adding points to Pk followed by adding lines to Lk. We assume that the original configuration is such that for every k ∈ N no two lines in Lk are parallel. Now we introduce some notation for this iterative process. The k stage is defined to consist of these two ordered steps: 1. Add each intersection of pairs of elements of Lk to Pk+1, and 2. Add a line through each of pair of elements of Pk to Lk+1.