Ramsey Theory, Set Theory, and Tukey Order

该研究计划涉及理论的发展，该理论基于拉姆齐理论，集合论和Tukey顺序的界面。 Ramsey理论是在给定的结构类中寻找规范结构的研究。 拓扑Ramsey空间是这样的拓扑空间，其中每个具有Baire性质的子集X都是Ramsey：空间中的每个非空开集都包含一个非空开子集，该子集包含在X中或与X不相交。 这样的空间统一了拉姆齐理论中的大量结果，包括卡尔森-辛普森定理、加尔文-普里克里定理、高尔斯定理、格雷厄姆-罗斯柴尔德定理、米利肯定理等。 最近的进展理论的拓扑拉姆齐空间由于Dobrinen和Todorcevic已被证明是有用的解决问题的集合论的精确分析之间的差异Tukey和Rudin-Keisler约化概念。 Tukey归约是Rudin-Keisler归约的一种弱化，它能够对其他方法无法分类的有序结构进行分类。 这个研究计划旨在发展一个一般的框架，规范等价关系的拓扑拉姆齐空间，并提供一个统一的理论，超滤满足一些弱划分性质。 这些新的Ramsey定理将被应用于分类的Tukey型超滤子的分割性质。 开发的方法旨在解决拉姆齐理论中一些长期存在的问题。 此外，本研究计划的目的是建立完整的Tukey结构的超滤波器，并找到模型理论的影响，Tukey订购的超功率。数学中的一个主要驱动力是在看似混乱中找到简单性。 拉姆齐理论是从大漩涡中寻找简单的规范结构的研究。 Tukey顺序是一种对对象强度进行分类的有用方法，从而通过将具有相同强度的所有对象分组在一起来简化对象的沼泽。 超滤子在数学的许多领域都很有趣，因为它们被用于构建数学理论的数学结构和模型。 特别是，超滤子是数学的基础，Tukey阶给出了超滤子强度的一个很好的概念。 该项目旨在开发解决拉姆齐理论中长期存在的开放问题的方法，同时显着提高我们对Tukey序和集合论的知识。 作为跨学科的，专注于基本问题，该项目预计将对数学的几个领域产生很大的影响。

This research program involves development of theory lying on the interface of Ramsey Theory, Set Theory, and Tukey Order. Ramsey theory is the study of finding canonical structures within a given class of structures. Topological Ramsey spaces are topological spaces in which every subset X which has the property of Baire is Ramsey: every non-empty open set in the space contains a non-empty open subset which is either contained in or disjoint from X. Such spaces unify a large body of results in Ramsey theory, including the theorems of Carlson-Simpson, Galvin-Prikry, Gowers, Graham-Rothschild, Milliken, and others. Recent progress in the theory of topological Ramsey spaces due to Dobrinen and Todorcevic has proved useful in solving problems in Set Theory regarding the precise analysis of the difference between the Tukey and Rudin-Keisler reducibility notions. Tukey reducibility, a weakening of Rudin-Keisler reducibility, is of high interest for its ability to classify ordered structures which defy classification by other means. This research program seeks to develop a general framework for canonical equivalence relations on topological Ramsey spaces and provide a unifying theory for ultrafilters satisfying some weak partition property. These new Ramsey theorems will be applied to classify the Tukey types of ultrafilters with partition properties. Methods developed are intended to solve some long-standing problems in Ramsey theory. Further, this research program aims to establish the full Tukey structure of ultrafilters and find the model-theoretic implications which Tukey ordering has for ultrapowers. A main driving force in mathematics is finding simplicity within seeming chaos. Ramsey theory is the study of finding simple, canonical structures from within a maelstrom. Tukey order is a useful means of classifying strengths of objects, thereby simplifying a morass of objects by grouping together all objects with the same strength. Ultrafilters are of interest in many areas of mathematics, as they are used for constructing mathematical structures and models of mathematical theories. In particular, ultrafilters are fundamental to mathematics, and Tukey order gives a good notion of the strength of an ultrafilter. This project aims to develop methods which will solve long-standing open problems in Ramsey theory while significantly improving our knowledge of Tukey Order and Set Theory. Being interdisciplinary and focused on fundamental problems, this project is expected to have high impact on several fields of mathematical.