A Bichromatic Incidence Bound and an Application

TL;DR

This paper establishes a new tight upper bound on point-hyperplane incidences in Euclidean space and applies it to derive combinatorial bounds on the number of spanned planes, also providing counterexamples to a known conjecture.

Contribution

It introduces a novel incidence bound in higher dimensions and uses it to derive new combinatorial results and counterexamples related to point configurations.

Findings

Proved a tight upper bound on point-hyperplane incidences in Euclidean d-space.

Derived a lower bound on the number of planes spanned by certain point sets.

Provided counterexamples to Purdy's conjecture in higher dimensions.

Abstract

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 O_d(m^{2/3}k^{2/3}n^{(d-2)/3} + kn^{d-2} + m) incidences between the k red points and m hyperplanes spanned by all n points provided that m = \Omega(n^{d-2}). For the monochromatic case k = n, this was proved by Agarwal and Aronov. 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 \Omega(nk^2) planes. We also provide an infinite family of counterexamples to a conjecture of Purdy's 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.