Ramsey Theory, Set Theory, and Tukey Order

拉姆齐理论、集合论和图基阶

• 批准号: 1600781
• 负责人: Natasha Dobrinen
• 金额: $ 13万
• 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: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600781&HistoricalAwards=false
• 关键词: Ramsey; Theory; Set; Tukey; Order

Ramsey theory is the study of finding order within seeming chaos. The classic example of this is Ramsey's Theorem, which states that for any coloring of all pairs of natural numbers into two colors, there is an infinite set of numbers from which every pair has the same color. Extensions of this theorem to more complex structures, rather than just pairs of numbers, have led to and continue to lead to breakthroughs in mathematics. The finding of exact copies of complex structures in which all small structures of some form behave simply is a means for sorting structural results in mathematics. Topological Ramsey space theory unifies many of the important theorems in the area into one general framework. The interrelations between in Ramsey theory, set theory, and Tukey order fuels progress in each of these areas. The project aims to continue development of topological Ramsey space theory and its applications to mapping exact structures in some fundamental topological spaces constructed from the natural numbers; to promote better understanding of the axiomatic foundations of mathematics; and to find dividing lines between those complex structures, for instance networks, which have large copies in which all small structures behave simply and those which do not. Ramsey theory and set theory overlap both in problems of interest and in methods of proof. This is seen in particular in the theory of topological Ramsey spaces, classic examples of which include the Ellentuck space, the Carlson-Simpson space of equivalence relations, and the Milliken space of infinite block sequences. Tukey reduction between partial orderings is a means for classifying partial orderings when isomorphism is too fine a notion to be useful. The Tukey structure of ultrafilters on the natural numbers is exactly the structure of the neighborhood bases in the Stone-Cech compactification of the natural numbers. The project's goals are to continue developing topological Ramsey space theory and its connections with set theory and Tukey structure, with applications to ultrahomogeneous relational structures and analysis. The aims of the project are several-fold but all interrelated. These include mapping the exact Tukey structure of ultrafilters on the natural numbers, and on Boolean algebras in general, and proving new pigeonhole principles relevant to the ultrafilters. In forcing theory, the project aims to streamline some areas of forcing by characterizing those partial orderings which are forcing equivalent to a topological Ramsey space, one particular focus being on creature forcings, again involving new pigeonhole principles. Another line of work is Ramsey theory at large cardinals, extending classical Ramsey theorems on countable structures to the uncountable. Finally, the project aims to solve problems regarding Ramsey theory on ultrahomogeneous relational structures, and to construct new types of Banach spaces via topological Ramsey space theory.

拉姆齐理论是在看似混乱的环境中寻找秩序的研究。这方面的经典例子是拉姆齐定理（Ramsey’s Theorem），该定理指出，对于将所有自然数对任意着色为两种颜色，存在一个无限的数集，其中每对自然数都具有相同的颜色。将这一定理扩展到更复杂的结构，而不仅仅是数字对，已经并将继续导致数学上的突破。寻找复杂结构的精确副本，其中所有某种形式的小结构都表现简单，是数学中对结构结果进行排序的一种手段。拓扑拉姆齐空间理论将该领域的许多重要定理统一到一个一般框架中。拉姆齐理论、集合理论和杜克秩序之间的相互关系推动了这些领域的进步。本项目旨在继续发展拓扑Ramsey空间理论及其在一些由自然数构成的基本拓扑空间中映射精确结构的应用；促进对数学公理基础的更好理解；并找到这些复杂结构之间的分界线，例如网络，在网络中，所有的小结构都表现得很简单，而在网络中，所有的小结构都表现得很简单。拉姆齐理论和集合论在兴趣问题和证明方法上都是重叠的。这在拓扑Ramsey空间理论中尤为明显，其经典例子包括Ellentuck空间、等价关系的Carlson-Simpson空间和无限块序列的Milliken空间。当同构概念过于精细而无法使用时，偏序间的Tukey约简是对偏序进行分类的一种方法。超过滤器在自然数上的土基结构正是自然数stone - ech紧化过程中邻基的结构。该项目的目标是继续发展拓扑Ramsey空间理论及其与集合论和Tukey结构的联系，并将其应用于超齐次关系结构和分析。该项目的目标是多方面的，但都是相互关联的。这些包括在自然数和布尔代数上映射超过滤器的精确Tukey结构，以及证明与超过滤器相关的新鸽子洞原理。在强迫理论中，该项目旨在通过描述那些与拓扑拉姆齐空间等价的偏序来简化强迫的某些领域，其中一个特别关注的是生物强迫，再次涉及新的鸽子洞原则。另一项工作是在大基数下的拉姆齐理论，将经典的可数结构拉姆齐定理扩展到不可数结构。最后，本课题旨在解决关于超齐次关系结构的Ramsey理论问题，并通过拓扑Ramsey空间理论构造新型的Banach空间。

项目成果

Continuous and other finitely generated canonical cofinal maps on ultrafilters

超滤器上的连续和其他有限生成的规范共最终图

• DOI: 10.4064/fm691-6-2019
• 发表时间: 2020
• 期刊: Fundamenta Mathematicae
• 影响因子: 0.6
• 作者: Dobrinen, Natasha

The Ramsey theory of the universal homogeneous triangle-free graph

• DOI: 10.1142/s0219061320500129
• 发表时间: 2017-04
• 期刊: J. Math. Log.
• 作者: Natasha Dobrinen

Perfect tree forcings for singular cardinals

奇异基数的完美树强迫

• 发表时间: 2020
• 期刊: Annals of pure and applied logic
• 影响因子: 0.8
• 作者: Dobrinen, Natasha;Hathaway, Dan;Prikry, Karel
• 通讯作者: Prikry, Karel