Vitaly Bergelson : Ergodic Ramsey Theory

**Abstract:** Our goal is provide a glimpse into the far reaching and mutually perpetuating connections between combinatorics and ergodic theory

The topics to be discussed:

- Early results and main principles of Ramsey theory;
- Partition Ramsey theory and topological dynamics;
- Density Ramsey theory and measure-preserving dynamics;
- Recent advancements and open problems

Neil Hindman : Ramsey Theory and the Stone-Čech Compactification

**Abstract:** If (S, •) is a discrete semigroup, the Stone-Čech compactification ßS of S can be viewed as the set of ultrafilters on S. Because of a crucial property of ultrafilters -- namely that whenever the union of finitely many sets is a member of the ultrafilter, one of them must also be a member -- ultrafilters are important in Ramsey Theory. Assuming only the knowledge of first year graduate courses in topology and algebra, I will develop enough of the algebraic theory of ßS to obtain some of its powerful applications to Ramsey Theory. Of particular importance is the theory of central sets. It is almost (but not quite) true that if P is an interesting property of subsets of a semigroup S, then every central set in S has property P. We shall examine many examples of this phenomenon.

Imre Leader : Ramsey Theory

**Abstract:** Ramsey theory asks the question: can we find some order in enough disorder? For example, suppose that we colour each natural number red or blue; can we guarantee to find an arithmetic progression of length 3 that is all one colour? The course will focus on Ramsey theory particularly in the context of the natural numbers. We will develop the cornerstones of the theory: Van der Waerden's theorem, which deals with arithmetic progressions, the Hales-Jewett theorem, which may be viewed as an abstract form of Van der Waerden's theorem, and Rado's theorem, which considers general linear structures.
No previous knowledge will be assumed (not even Ramsey's theorem). Along the
way, we will mention several open problems.