# Mini-course “Infinitary Ramsey Theory”, part I: Topological Ramsey spaces and applications to ultrafilters

Mini-course given by Natasha Dobrinen (University of Denver, Colorado, USA) at the KGRC seminar on 2019‑01‑08. (See also part II.)

### Abstract

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
*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.

There is a video recording of this course on YouTube.

The slides for this talk are available, too.

### Time and Place

Course at 10:30am in the KGRC lecture room