From Geometry to Combinatorics and Back: Escaping the Curse of Dimensionality

An ERC Advanced Grant
September 2020 - August 2025

The main goal of the present project is to attack some hard problems for large classes of graphs and hypergraphs arising in geometric, algebraic, and practical applications. These structures escape the ""curse of dimensionality”: they can be embedded in a bounded-dimensional space, or they have small VC-dimension, or a short algebraic description. The work of the principal investigator, his collaborators and students has played a significant role in the introduction of modern combinatorial tools in geometry. In the present project, he aims to explore the reverse direction: to develop and apply geometric techniques to settle important special cases of notoriously difficult combinatorial problems on (1) bounded degree semi-algebraic graphs and hypergraphs, (2) graphs and hypergraph of bounded VC-dimension, (3) ordered graphs, 0-1 matrices, and graphs embedded in the plane or in other surfaces. Progress on the problems described in the proposal is expected to lead closer to the solution of some classical problems such as the Erdős-Hajnal conjecture, the Danzer-Rogers conjecture, the Schur-Erdős problem, and to the development of improved algorithms for clustering and property testing in huge graphs.