Extremal Combinatorics and Ramsey Theory in Structured Settings

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This research project focuses broadly on the area of combinatorics, and the questions under consideration are characterized by their connections to several different fields of mathematics. One of the main aims of this project is to explore these connections, and to synthesize techniques from different areas and bring them to bear on some longstanding open problems. Much of the planned work is concerned with the properties of large discrete structures, and the questions of interest in this project are relevant to the study of disordered systems that may be modeled by such structures, such as collections of interacting particles, large networks like the internet, or the human brain.There are two primary areas under investigation, namely extremal combinatorics and Ramsey theory. The project will investigate questions in extremal combinatorics that involve additional structure, with a focus on algebraic, number-theoretic, and topological structure. The project will also address questions of a Ramsey-theoretic nature that additionally have additive-combinatorial, algebraic, and ergodic-theoretic connections. Finally, the project will pursue various connections between problems in these areas. Specific questions to be attacked include a topological Dirac-type theorem for hypergraphs, investigating product-free sets living inside free structures, improving bounds for a few Ramsey-theoretic results of an additive-combinatorial flavor, and investigating the Ramsey-theoretic properties of non-abelian groups.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.