Logic, Ramsey Theory, and Relational Structures

Ramsey Theory is a central area of mathematics aptly characterized by Motzkin's motto, "Complete disorder is impossible." It is often the case that by starting with a large enough structure, a substructure with desired properties emerges. Ramsey's Theorem states that given any coloring of all pairs of natural numbers into finitely many colors, there is an infinite subset in which all pairs have the same color. Since its inception, Ramsey theory has developed in multiple directions, often appearing as the core content in solutions to deep problems from a wide range of mathematical disciplines. This project utilizes techniques in mathematical logic to more fully develop Ramsey theory of infinite relational structures. A major motivation is to find dividing lines between those infinite structures which act like the natural numbers in the sense of possessing analogues of Ramsey's theorem, and those which do not. Progress on infinite structures works in tandem with progress in mathematical logic and topology, creating new pathways between several areas of mathematics. This project includes important questions suitable for graduate students and early career researchers, thus providing opportunities to broaden participation of well-trained mathematicians via the PI's mentoring. This research program investigates infinite relational structures supporting Ramsey theory. A major focus of this project is to develop the Ramsey theory of homogeneous structures. This involves constructing new types of trees which code homogeneous relational structures and using the technique of forcing to produce (in ZFC) Ramsey theorems for these classes of trees, as well as developing purely combinatorial proofs. Computability theoretic strengths of structural Ramsey statements will be investigated, as will model-theoretic dividing lines. The second main focus is the continued development of topological Ramsey space theory and its implications for forcing, ultrafilters, and Banach spaces. Development of Ramsey spaces for homogeneous structures ties this together with the first focus. The third line of research will develop Ramsey theory on uncountable structures. The techniques developed, involving simultaneous uses of logic, combinatorics and topology, will create new pathways between these areas of mathematics. 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.