Time and Place: Tuesday, January 8 and Wednesday, January 9 at 10:30am in the KGRC lecture room (both parts) at the KGRC.

**Part I. Topological Ramsey spaces and applications to ultrafilters **

**Part II. Ramsey theory on trees and applications to big Ramsey degrees**

The Infinite Ramsey Theorem states that given $n,r\ge 1$ and a coloring of

all $n$-sized subsets of $\mathbb{N}$ into $r$ colors, there is an

infinite subset of $\mathbb{N}$ in which all $n$-sized subsets have the

same color. There are several natural ways of extending Ramsey’s Theorem.

One extension is to color infinite sets rather than finite sets. In this

case, the Axiom of Choice precludes a full-fledged generalization, but

upon restricting to definable colorings, much can still be said. Another

way to extend Ramsey’s Theorem is to color finite sub-objects of an

infinite structure, requiring an infinite substructure isomorphic to the

original one. While it is not possible in general to obtain substructures

on which the coloring is monochromatic, sometimes one can find bounds on

the number of colors, and this can have implications in topological

dynamics.

In Part I, we will trace the development of Ramsey theory on the Baire

space, from the Nash-Williams Theorem for colorings of clopen sets to the

Galvin-Prikry Theorem for Borel colorings, culminating in Ellentuck’s

Theorem correlating the Ramsey property with the property of Baire in a

topology refining the metric topology on the Baire space. This refinement

is called the Ellentuck topology and is closely connected with Mathias

forcing. Several classical spaces with similar properties will be

presented, including the Carlson-Simpson space and the Milliken space of

block sequences. From these we shall derive the key properties of

topological Ramsey spaces, first abstracted by Carlson and Simpson and

more recently given a refined presentation by Todorcevic in his book {\em

Introduction to Ramsey spaces}. As the Mathias forcing is closely

connected with Ramsey ultrafilters, via forcing mod finite initial

segments, so too any Ramsey space has a $\sigma$-closed version which

forces an ultrafilter with partition properties. Part I will show how

Ramsey spaces can be used to find general schemata into which disparate

results on ultrafilters can be seen as special cases, as well as obtain

fine-tuned results for structures involving ultrafilters.

Part II will focus on Ramsey theory on trees and their applications to

Ramsey theory of homogeneous structures. An infinite structure is {\em

homogeneous} if each isomorphism between two finite substructures can be

extended to an automorphism of the infinite structure. The rationals as a

linearly ordered structure and the Rado graph are prime examples of

homogeneous structures. Given a coloring of singletons in the rationals,

one can find a subset isomorphic to the rationals in which all singletons

have the same color. However, when one colors pairs of rationals, there

is a coloring due to Sierpinski for which any subset isomorphic to the

rationals has more than one color on its pairsets. This is the origin of

the theory of {\em big Ramsey degrees}, a term coined by Kechris, Pestov

and Todorcevic, which investigates bounds on colorings of finite

structures inside infinite structures. Somewhat surprisingly, a theorem

of Halpern and L\”{a}uchli involves colorings of products of trees,

discovered en route to a proof that the Boolean Prime Ideal Theorem is

strictly weaker than the Axiom of Choice, is the heart of most results on

big Ramsey degrees. We will survey big Ramsey degree results on countable

and uncountable structures and related Ramsey theorems on trees, including

various results of Dobrinen, Devlin, D\v{z}amonja, Hathaway, Larson,

Laver, Mitchell, Shelah, and Zhang.