Generalizations of the Szemerédi-Trotter Theorem

In this paper, we generalize the Szemerédi-Trotter theorem, a fundamental result of incidence geometry in the plane, to flags in higher dimensions. In particular, we employ a stronger version of the polynomial cell decomposition technique, which has recently shown to be a powerful tool, to generalize the Szemerédi-Trotter Theorem to an upper bound for the number of incidences of complete flags in Rn (i.e. amongst sets of points, lines, planes, etc.). We also consider variants of this problem in three dimensions, such as the incidences of points and light-like lines, as well as the incidences of points, lines, and planes, where the number of points and planes on each line is restricted. Finally, we explore a group theoretic interpretation of flags, which leads us to new incidence problems.