Model Theoretic Classification Theory and Finite Combinatorics

模型论是数学逻辑的一个分支，它试图理解驱动不同类型数学对象行为的常见结构现象。 这一领域的一个关键思想是分界线的概念，它是由Shelah在20世纪70年代首先提出的。分界线可以被认为是某类数学对象中的结构二分法。 许多最重要的分界线对应于局部组合性质，这些性质对全局结构具有重要意义。 在无限的情况下，模型理论家已经成功地使用分界线对例子进行分类并概括它们的行为。然而，扩展到有限的设置一直是有限的，主要是由于失败的关键无限的工具。 另一方面，极值和算术组合学是专注于有限设置的领域，但它们研究许多与模型理论相同的主题，例如局部与全局结构以及结构和随机性的相互作用。 这些领域发展了有限的问题和工具，这些问题和工具对模型论来说是新的，但与模型论思想有着深刻的联系。 本项目的目标是通过解决极值和加法组合学中的有限问题来扩展有限环境中模型理论分界线的研究，更具体地说，本项目将专注于寻找对加法和极值组合学中的定理中的边界和增长率具有鲁棒意义的局部模型理论条件。 这将在两个主要方向上实现。 第一个将集中在加法组合学的问题。 这里的一个主要目标将是确定结构二分法的子集的高维向量空间在素域。例如，什么样的集合是最“驯服”的，通过结构分解定理中的改进边界来测量？ “驯服”集能用局部组合构形来刻画吗？ 第二个方向将解决极值组合学中的问题。 具体来说，PI将继续致力于有限关系语言中遗传属性的枚举和极值问题。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.