Logic, Ramsey Theory, and Relational Structures

拉姆齐理论是数学的核心领域，恰如其分地体现了莫茨金的格言“完全无序是不可能的”。拉姆齐定理指出，给定所有自然数对的任意着色为有限多种颜色，存在一个无限子集，其中所有自然数对都具有相同的颜色。拉姆齐理论自提出以来，已向多个方向发展，经常作为解决各种数学学科深层问题的核心内容出现。本项目利用数理逻辑技术，更充分地发展了无限关系结构的拉姆齐理论。一个主要的动机是在那些像自然数一样的无限结构和那些不像自然数的无限结构之间找到分界线，这些结构具有拉姆齐定理的类似物。特别令人感兴趣的是拉姆齐理论在禁止构型结构中的进步。无限结构的进展与数学逻辑和拓扑学的进展是同步的，在数学的几个领域之间创造了新的途径。该项目包括一些适合研究生和早期职业研究人员的重要问题，从而为通过PI的指导扩大训练有素的数学家的参与提供了机会。这个研究项目将发展无限关系结构的拉姆齐理论，特别是那些具有禁止构型的结构，在PI最近解决普遍齐次无三角形图之前，这一领域在很大程度上是不受研究影响的。这将涉及构建编码同质关系结构的新类型的树，并使用强制生成（在ZFC中）这些树的拉姆齐定理的技术。这些新技术将为有限结构Ramsey理论提供更好的边界。本文将研究变Ramsey语句的可计算性理论强度，以及与模型理论中分类方案的联系。无限拉姆齐理论将继续发展与广泛的含义和应用到其他领域的数学。所开发的技术，包括同时使用逻辑学、组合学和拓扑学，将在这些数学领域之间创造新的途径。该奖项反映了美国国家科学基金会的法定使命，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。

Ramsey Theory is a central area of mathematics aptly characterized by Motzkin's motto, "Complete disorder is impossible." 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. Of particular interest is the advancement of Ramsey theory for structures with forbidden configurations. 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 some important questions which are 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 will develop the Ramsey theory of infinite relational structures, especially those with forbidden configurations, an area which had been largely impervious to investigations prior to the PI's recent solution for the universal homogeneous triangle-free graph. This will involve 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. These new techniques will be used obtain better bounds for finite structural Ramsey theory. Computability theoretic strengths of varying Ramsey statements, and connections with classification schemes in model theory will be investigated. Infinitary Ramsey theory will continue to be developed with a broad spectrum of implications for and applications to other areas of mathematics. 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.