Logic, Ramsey Theory, and Relational Structures

拉姆齐理论是数学的核心领域，莫茨金的座右铭是：“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颜色为200多种颜色，存在一个无限子集，其中所有对都具有相同的颜色。 自诞生以来，拉姆齐理论已经向多个方向发展，经常作为核心内容出现在解决广泛的数学学科的深层问题中。 该项目利用数理逻辑中的技术来更充分地发展无限关系结构的拉姆齐理论。 一个主要的动机是找到那些无限结构之间的分界线，这些无限结构在拥有拉姆齐定理类似物的意义上表现得像自然数，而那些不像自然数。 特别令人感兴趣的是拉姆齐理论的进步与禁止配置的结构。 无限结构的进展与数学逻辑和拓扑学的进展相结合，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，从而提供机会，通过PI的指导扩大训练有素的数学家的参与。该研究计划将发展无限关系结构的拉姆齐理论，特别是那些具有禁止配置的理论，在PI最近解决了通用齐次无三角形图之前，该领域在很大程度上没有受到调查的影响。 这将涉及构建编码同构关系结构的新类型的树，并使用强制产生（在ZFC中）这些树类的拉姆齐定理的技术。 这些新的技巧将被用来获得有限结构Ramsey理论的更好的界。 可计算性理论的优势，不同的拉姆齐声明，并与模型理论中的分类方案的连接将进行调查。 无穷拉姆齐理论将继续发展与广泛的影响和应用到其他领域的数学。 所开发的技术，涉及逻辑，组合学和拓扑学的同时使用，将创建这些数学领域之间的新途径。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.