Ramsey Theory, Set Theory, and Tukey Order

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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.