On geometric versions of Zarankiewicz’s problem

Extremal combinatorics poses a fundamental question: How large can a system be while avoiding certain configurations? A classic instance of this inquiry arises in extremal graph theory: Given a fixed graph H, what is the maximum number ex(n, H) of edges a graph G on n vertices can have if it excludes H as a subgraph? This problem remains widely open for H being a complete bipartite graph and is known as Zarankiewicz’s problem.

Even when considering algebraic constraints on the hosting graph G, such as being the incidence graph of points and bi-variate polynomials of fixed degree, Zarankiewicz’s problem remains notoriously challenging. This geometric interpretation of Zarankiewicz’s problem has led to the emergence of Incidence Geometry.

In this talk, I will provide an overview of notable results in this domain and will introduce a novel approach to Zarankiewicz’s problem. 

Based on joint work with Chaya Keller.