A semi-algebraic version of Zarankiewicz's problem

Abstract

A bipartite graph G is semi-algebraic in R^d if its vertices are represented by point sets P,Q ⊂ R^d and its edges are defined as pairs of points (p,q) ∈ P×Q that satisfy a Boolean combination of a fixed number of polynomial equations and inequalities in 2d coordinates. We show that for fixed k, the maximum number of edges in a K_(k,k)-free semi-algebraic bipartite graph G=(P,Q,E) in R^2 with |P|=m and |Q|=n is at most O((mn)^(2/3) + m + n), and this bound is tight. In dimensions d ≥ 3, we show that all such semi-algebraic graphs have at most C((mn)^(dd+1+ϵ) + m + n) edges, where here ϵ is an arbitrarily small constant and C=C(d,k,t,ϵ). This result is a far-reaching generalization of the classical Szemeredi-Trotter incidence theorem. The proof combines tools from several fields: VC-dimension and shatter functions, polynomial partitioning, and Hilbert polynomials. We also present various applications of our theorem. For example, a general point-variety incidence bound in R^d, an improved bound for a d-dimensional variant of the Erdos unit distances problem, and more.