RUI: Model Theory and Structural Ramsey Theory

In a large enough data set, a certain amount of uniformity is guaranteed. Variations on this theme are called "Ramsey theorems" after the groundbreaking work of Frank Ramsey in the early 20th century. Ramsey theorems have been used most recently in computer science to design efficient algorithms on certain types of data sets. This project concerns structural Ramsey theory, which uses tools from model theory to describe the objects under study. Model theory is an area of foundations of mathematics that studies properties holding generally on large sets of mathematical objects, across different areas of mathematics. This research project will extend the contribution of model theory to structural Ramsey theory. Methods from saturated model theory, pseudofinite model theory, category theory, and topological dynamics will be applied to create new knowledge pathways across these different areas of mathematics. This project will also provide increased training opportunities for students at California State University, San Bernardino. This research project studies how certain maps between two infinite structures in possibly different first-order languages can transfer information about the automorphism groups of these structures and, ultimately, the Ramsey properties of the finitely-generated structures embeddable in these infinite structures. In particular, this project studies a pair of maps between two infinite structures called a "semi-retraction" that induces a retraction of the type space of one structure onto the type space of the other. This project also studies which Ramsey theorems transfer from a class of finite structures to ultraproducts of these finite structures. This project will identify notions of tame partitions and investigate the classes of structures which have a Ramsey theorem for these tame partitions, but perhaps not in general. Many test cases are available as a result of a confluence of recent results in the calculation of finite big Ramsey degrees. This work is anticipated to have applications in classification theory in model theory, as well as in topological dynamics. 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.