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。“通常情况下，通过从足够大的结构开始，具有所需特性的子结构就会出现。 拉姆齐定理指出，给定所有自然数对的任意着色为100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。 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。该项目包括适合研究生和早期职业研究人员的重要问题，从而通过PI的指导提供了扩大训练有素的数学家参与的机会。该研究计划调查支持拉姆齐理论的无限关系结构。 该项目的一个主要重点是发展拉姆齐均匀结构理论。 这涉及到构建新类型的树，这些树编码同构关系结构，并使用强制产生（在ZFC中）拉姆齐定理的技术来证明这些树类，以及开发纯粹的组合证明。 结构拉姆齐语句的可计算性理论的优势将进行调查，将模型理论的分界线。 第二个主要重点是继续发展的拓扑拉姆齐空间理论和它的影响，迫使，超过滤器和Banach空间。 齐次结构的Ramsey空间的发展将这与第一个焦点联系在一起。 第三条研究路线将发展拉姆齐理论的不可数结构。 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." 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.