Ramsey Theory, Set Theory, and Tukey Order

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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.