Model Theoretic Classification Theory and Finite Combinatorics

模型理论是数理逻辑的一个分支，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想法是分界线的概念，该想法最早由谢拉在20世纪70年代提出。分界线可以被认为是某类数学对象中的结构性二分法。许多最重要的分界线对应于对全球结构具有重大影响的局部组合性质。在无限的背景下，模型理论家们已经取得了巨大的成功，他们使用分界线来对例子进行分类并概括它们的行为。然而，对有限环境的扩展一直是有限的，主要是因为关键的无限工具在那里失败了。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，如局部结构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理论来说是新的，但与模型理论的思想有着深刻的联系。这个项目的目标是通过解决极值和加性组合学中的有限问题来扩展有限环境下模型理论分界线的研究。更具体地说，这个项目将专注于寻找局部模型理论条件，这些条件对于加性和极值组合学中的定理中的界和增长率具有稳健的含义。这将在两个主要方向上实现。第一个问题将集中在加法组合学的问题上。这里的一个主要目标将是确定素数域上高维向量空间的子集的结构二分法。例如，通过结构分解定理中改进的界限来衡量，哪种类型的集合是最“驯服”的？“驯服”集合可以用局部组合构型来刻画吗？第二个方向将解决极值组合数学中的问题。具体地说，PI将继续研究有限关系语言中遗传属性的枚举和极值问题。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.