Geometry Seminar

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.