Model Theoretic Classification Theory and Finite Combinatorics

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Model theory is a branch of mathematical logic which seeks to understand common structural phenomena driving the behavior of different types of mathematical objects. A crucial idea in this area, first developed in the 1970s by Shelah, is the notion of a dividing line. A dividing line can be thought of as a structural dichotomy within a certain class of mathematical objects. Many of the most important dividing lines correspond to local combinatorial properties which have significant implications for global structure. In the infinite setting, model theorists have had great success using dividing lines to classify examples and generalize their behavior. However, extensions into the finite setting have been limited, largely due to the failure there of crucial infinitary tools. On the other hand, extremal and arithmetic combinatorics are fields which focus on the finite setting, but which study many of the same themes as model theory, such as local versus global structure and the interplay of structure and randomness. These fields have developed finitary questions and tools which are new to model theory, but which have have deep connections to model theoretic ideas. The goal of this project is to extend the study of model theoretic dividing lines in the finite setting by solving finitary problems from extremal and additive combinatorics which address these shared themes.More specifically, this project will focus on finding local model theoretic conditions which have robust implications for bounds and growth rates in theorems from additive and extremal combinatorics. This will be accomplished in two main directions. The first will focus on questions from additive combinatorics. Here a main goal will be to identify structural dichotomies for subsets of high-dimensional vector spaces over prime fields. For instance, what kinds of sets are most "tame'', as measured through improved bounds in structural decomposition theorems? Can the "tame'' sets be characterized by local combinatorial configurations? The second direction will address questions in extremal combinatorics. Specifically, the PI will continue work on enumeration and extremal problems for hereditary properties in finite relational languages.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.