Logic, Ramsey Theory, and Relational Structures

拉姆齐理论是数学的中心领域，莫茨金的座右铭“完全无序是不可能的”恰如其分地体现了这一点。拉姆齐定理指出，给定所有自然数对的任何颜色为有限多个颜色，则存在一个无限子集，其中所有的自然数对具有相同的颜色。自创立以来，拉姆齐理论已经向多个方向发展，经常作为解决来自广泛数学学科的深层次问题的核心内容出现。这个项目利用数理逻辑的技术来更全面地发展无限关系结构的拉姆齐理论。一个主要的动机是在那些在具有拉姆齐定理类似物的意义上表现为自然数的无限结构和那些不具有类似物的无限结构之间找到分界线。特别令人感兴趣的是拉姆齐理论对具有禁止构型的结构的改进。无限结构的进步与数理逻辑和拓扑学的进步齐头并进，在数学的几个领域之间创造了新的途径。这个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，从而提供了通过PI的指导来扩大训练有素的数学家参与的机会。本研究计划将发展无限关系结构的Ramsey理论，特别是那些具有禁止配置的结构，在PI最近解决通用齐次无三角形图之前，这个领域在很大程度上不受调查的影响。这将涉及构建编码同构关系结构的新类型的树，并使用强制技术来为这些类树产生(在ZFC中)Ramsey定理。这些新技术将被用来获得有限结构Ramsey理论的更好界。我们将研究各种Ramsey语句的可计算性理论优势，以及与模型理论中的分类方案之间的联系。无限拉姆齐理论将继续得到发展，并对数学的其他领域产生广泛的影响和应用。开发的技术涉及同时使用逻辑、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。

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.