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(m2/3k2/3n(d−2)/3+knd−2+m) incidences between the k red points and m hyperplanes spanned by all n points provided that m=Ω(nd−2). For the monochromatic case k=n, this was proved by Agarwal and Aronov (Discrete Comput. Geom. 7(4):359–369, 1992).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(nk2) planes. We also provide an infinite family of counterexamples to a conjecture of Purdy’s (Erdős and Purdy in Handbook of Combinatorics, vol. 1, pp. 809–873, Elsevier, Amsterdam, 1995) 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.

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