On Incidences Between Points and Hyperplanes

Abstract

We show that if the number I of incidences between m points and n planes in R³ is sufficiently large, then the incidence graph (that connects points to their incident planes) contains a large complete bipartite subgraph involving r points and s planes, so that rs ≥ I2 mn −a(m+n), for some constant a> 0. This is shown to be almost tight in the worst case because there are examples of arbitrarily large sets of points and planes where the largest complete bipartite incidence subgraph records only I2 m+n mn − 16 incidences. We also make some steps towards generalizing this result to higher dimensions. On the way, we slightly improve upon a result of Brass and Knauer [BK] about the representation complexity of incidences between m points and n hyperplanes in Rd, and get rid of the logarithmic factor in their upper bound