Logic, Ramsey Theory, and Relational Structures

逻辑、拉姆齐理论和关系结构

• 批准号: 1901753
• 负责人: Natasha Dobrinen
• 金额: $ 15.85万
• 依托单位: University of Denver
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-09-01 至 2022-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1901753&HistoricalAwards=false
• 关键词: Logic; Ramsey; Theory; Relational; Structures

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.

拉姆齐理论是数学的核心领域，莫茨金的座右铭“完全无序是不可能的”恰如其分地描述了这一点。 拉姆齐定理指出，给定所有自然数对的任意颜色为有限多种颜色，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，拉姆齐理论向多个方向发展，经常作为解决广泛数学学科的深层问题的核心内容出现。 该项目利用数理逻辑技术来更充分地发展拉姆齐无限关系结构理论。 一个主要动机是找到那些像自然数一样具有拉姆齐定理类似物的无限结构和不具有拉姆齐定理类似物的无限结构之间的分界线。 特别令人感兴趣的是拉姆齐禁构结构理论的进展。 无限结构的进步与数理逻辑和拓扑学的进步协同作用，在多个数学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，从而为通过 PI 的指导扩大训练有素的数学家的参与提供了机会。该研究计划将发展拉姆齐无限关系结构理论，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，这一领域基本上不受研究影响。 这将涉及构建编码同质关系结构的新型树，并使用强制技术为这些树类生成（在 ZFC 中）拉姆齐定理。 这些新技术将用于为有限结构拉姆齐理论获得更好的界限。 将研究不同拉姆齐陈述的可计算性理论优势，以及与模型理论中分类方案的联系。 无穷拉姆齐理论将继续发展，对数学的其他领域具有广泛的影响和应用。 所开发的技术涉及逻辑、组合学和拓扑的同时使用，将在这些数学领域之间创建新的途径。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Forcing and the Halpern-Lauchli Theorem

强迫和 Halpern-Lauchli 定理

• DOI: 10.1017/jsl.2019.59
• 发表时间: 2020
• 期刊: The journal of symbolic logic
• 作者: Dobrinen, Natasha;Hathaway, Daniel
• 通讯作者: Hathaway, Daniel