Natasha Dobrinen: Mini-course on Infinitary Ramsey theory

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.

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