A Bichromatic Incidence Bound and an Application

We prove a new, tight upper bound on the number of incidences between points and hyperplanes in Euclidean d-space. Given n points, of which k are colored red, there are Od(m 2/3 k 2/3 n +kn+m) incidences between the k red points and m hyperplanes spanned by all n points provided that m = Ω(n). For the monochromatic case k = n, this was proved by Agarwal and Aronov [1]. We use this incidence bound to prove that a set of n points, no more than n − k of which lie on any plane or two lines, spans Ω(nk) planes. We also provide an infinite family of counterexamples to a conjecture of Purdy’s [2] on the number of hyperplanes spanned by a set of points in dimensions higher than 3, and present new conjectures not subject to the counterexample.