Research in Extremal Combinatorics

Extremal combinatorics has grown significantly in both depth and breadth in the 21st century, resulting in methods that apply well beyond their original settings. These include applications, and often significant breakthroughs, in number theory, design theory, probability, group theory, and theoretical computer science. The questions that are studied here represent obstacles to our understanding, and it is likely that progress on these challenges and the methods developed in solving them can later be exported to other areas.

As such, a major goal of this project is to develop new tools and techniques and to expand on existing methods so that they apply in more general settings. This work will involve training undergraduate and graduate students.

The PI will study a variety of fundamental questions in extremal combinatorics, spanning three main areas: extremal graph theory, Ramsey theory, and the study of pseudorandomness. In the first area, the PI intends to study several classical questions on extremal numbers, including the rational exponents conjecture and the compactness conjecture, as well as problems regarding the relationships between the homomorphism densities of different graphs.

In the second area, the PI will tackle some of the classical problems on graph Ramsey theory, such as that of determining diagonal Ramsey numbers, as well as questions from arithmetic and geometric Ramsey theory. Finally, the PI intends to make further progress on developing the sparse regularity method and on finding and applying high-dimensional expanders.

Common themes run through these areas, and the methods developed in one area are likely to have implications for the others.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.