Daniel Varga

A general framework for bounding the independence number of geometric hypergraphs, Part I: The Method
A general framework for bounding the independence number of geometric hypergraphs, Part II: Applications

We present the Geometric Fractional Chromatic Number (GFCN), a method for upper-bounding the independence number of geometric graphs and hypergraphs. The material is split across two talks. The first introduces the method, relying more on computer animations than formulas. The second surveys a range of applications, from geometric graph theory to number theory. No background beyond basic undergraduate mathematics is assumed for either talk.

The method was first applied to bound the density of unit distance-avoiding sets in the plane, answering a question posed by Paul Erdős. Since then, it has led to several further results, including:

- constructing unit-distance graphs with independence ratio arbitrarily close to 1/4, making progress on another problem of Erdős;
- a new bound on the density of orthogonality-avoiding sets on the sphere;
- a number-theoretic application, about subsets $A \subset F_p$ such that $A+A-2A \neq F_p$.