Caltech/UCLA/USC Joint Analysis Seminar

The symmetric case of the Szemer\'edi-Trotter theorem says that any configuration of N lines and N points in the plane has at most O(N^{4/3}) incidences. We describe a recipe involving just O(N^{1/3}) parameters which sometimes (that is, for some choices of the parameters) produces a configuration of N point and N lines. (Otherwise, we say the recipe fails.) We show that any near-extremal example for Szemer\'edi Trotter is densely related to a successful instance of the recipe. We discuss the relation of this statement to the inverse Szemer\'edi Trotter problem. )