The Largest Complete Bipartite Subgraph in Point-Hyperplane Incidence Graphs

Given m points and n hyperplanes in R-d (d >= 3), if there are many incidences, we expect to find a big cluster K-r,K-s in their incidence graph. Apfelbaum and Sharir [1] found lower and upper bounds for the largest size of rs, which match (up to a constant) only in three dimensions. In this paper we close the gap in four and five dimensions, up to some polylogarithmic factors.