CAREER: Extremal Combinatorics: Methods, Problems, and Challenges

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This research project considers a variety of problems related to Szemerédi's regularity method and Ramsey theory. In tackling these problems, the PI will use a range of combinatorial methods that have recently led to substantial progress on related problems. Examples include probabilistic methods, density increment arguments, transference arguments, analytic tools, and embedding techniques. The first area in this project concerns Szemerédi's regularity method. Within this area, one of the main goals of the project is to obtain new bounds on the triangle removal lemma and its various extensions and variants. The triangle removal lemma states that any graph with a subcubic number of triangles can be made triangle-free by removing a subquadratic number of edges. Another major goal of the project is to further push the regularity method to sparse graphs and other combinatorial structures, and to obtain new applications. Specific problems include optimizing the pseudorandomness conditions needed to obtain sparse counting lemmas, proving analogous sparse regularity results in other combinatorial structures such as cubes, and providing new applications in number theory and discrete geometry such as extensions of the Green-Tao theorem on long arithmetic progressions in the primes. The second area in this project is estimating Ramsey numbers. The PI will work on proving new bounds for classical (complete) graph and hypergraph Ramsey numbers, and to prove linear bounds for Ramsey numbers of sparse graphs.This project studies fundamental problems in combinatorics related to the structure of large networks. Examples of large networks include the Internet, Facebook, the brain, imperfect crystals, and designed chips. The structure of these networks can be critical in understanding how the networks function. Previous work has shown that the subjects under study in this project have a wide range of applications. Furthermore, this work has led to the development of powerful methods that have been used in many branches of mathematics and computer science. For example, previous progress on estimating Ramsey numbers led to the development of probabilistic techniques that have had a tremendous influence on computer science, such as in the design of randomized algorithms. It is expected that further work on these problems will lead to new methods and applications.