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。通常的情况是，从一个足够大的结构开始，就会出现一个具有所需特性的子结构。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。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。本研究项目调查支持拉姆齐理论的无限关系结构。这个项目的一个主要焦点是发展均匀结构的拉姆齐理论。这包括构造编码同构关系结构的新型树，使用强制技术为这些类树产生(在ZFC中)Ramsey定理，以及开发纯粹的组合证明。我们将研究结构Ramsey语句的可计算性理论优势，以及模型理论的分界线。第二个主要焦点是拓扑拉姆齐空间理论的继续发展及其对强迫空间、超滤子空间和Banach空间的影响。齐次结构的Ramsey空间的发展将这一点与第一个焦点联系在一起。第三条研究线将发展拉姆齐关于不可数结构的理论。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." It is often the case that by starting with a large enough structure, a substructure with desired properties emerges. 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. 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 important questions 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 investigates infinite relational structures supporting Ramsey theory. A major focus of this project is to develop the Ramsey theory of homogeneous structures. This involves 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, as well as developing purely combinatorial proofs. Computability theoretic strengths of structural Ramsey statements will be investigated, as will model-theoretic dividing lines. The second main focus is the continued development of topological Ramsey space theory and its implications for forcing, ultrafilters, and Banach spaces. Development of Ramsey spaces for homogeneous structures ties this together with the first focus. The third line of research will develop Ramsey theory on uncountable structures. 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.