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”。通常情况下，从一个足够大的结构开始，一个具有所需性质的子结构就会出现。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。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机会。本研究计划探讨支持拉姆齐理论的无限关系结构。本项目的主要重点是发展拉姆齐同质结构理论。这包括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，以及开发纯粹的组合证明。本文将研究结构Ramsey语句的可计算性理论强度，以及模型理论分界线。第二个主要焦点是拓扑Ramsey空间理论的持续发展及其对强迫、超滤和巴拿赫空间的影响。齐质结构的拉姆齐空间的发展将这与第一个重点联系在一起。第三条研究线将发展不可数结构的拉姆齐理论。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." 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.