MATH 7810: Logic Seminar: Applications of infinitary Ramsey theory (Fall 2010)
Instructor: Justin Moore
If X is a set and we partition the pairs of elements of X into two pieces, is there a large subset of X, all of whose pairs belong to a single piece? This is the basic question in Ramsey theory which can be varied in any number of ways: by increasing "pairs" to "triples" or even infinite sequences; by placing additional restrictions on the partition; by weakening the homogeneity requirement. There are a large variety of example of problems arising outside of set theory in which the core essence of the problem involves Ramsey theory where either the dimension is infinite or else the underlying set X is uncountable.
This seminar will focus on the Ramsey theory of infinite sets and how it arises naturally through classification problems in set theory and analysis. The starting point for this seminar will be a seminal paper by Todorcevic in which he proves a number of classification results for compact subsets of the Baire class 1 functions. On one hand, it utilizes strengthenings of the infinite dimensional Ramsey theorem of Galvin and Prikry. On the other hand, the dichotomies proved in the paper are motivated by the study of finite dimensional Ramsey theory on uncountable sets using additional axioms of set theory such as forcing axioms.
Participants in the course should have taken a graduate level course in real analysis and will benefit from prior exposure to set theory (although with some motivation, this is not necessary). Those enrolled in the course are expected to present material related to the topic of the seminar.