Natasha Dobrinen, University of Notre Dame
"Infinite-dimensional Ramsey theory on binary relational homogeneous structures"
The infinite pigeonhole principle states that given a coloring of the natural numbers into finitely many colors, there are infinitely many numbers with the same color. Ramsey's celebrated theorem extended this to colorings of k-sized sets for any positive integer k. For instance, given any coloring of all pairs of natural numbers into finitely many colors, there is an infinite set X of natural numbers so that all pairs of numbers in X have the same color. This fact has a structural interpretation: Given any finite coloring of the edges of a complete graph on infinitely many vertices, there is an infinite complete subgraph in which all the edges have the same color.
When one moves from coloring finite sets of a fixed size to finite sets of different sizes or even coloring infinite sets, new constraints have to be added. Amazingly, these constraints can be viewed topologically. The Baire space is the topological space of all infinite subsets of the natural numbers. Due the Axiom of Choice one cannot hope to obtain a direct analogue of Ramsey's theorem for colorings of the Baire space. However, a series of fascinating results in the 1960s and 1970s showed culminated in Ellentuck's topological characterization, using the Vietoris topology, of those colorings for which an infinite-dimensional Ramsey theorem holds.
Structural Ramsey theory is concerned with coloring copies of a given structure inside a large ambient structure. In the seminal 2005 paper of Kechris-Pestov-Todorcevic, a series of questions were asked about developing Ramsey theory on infinite structures. We will outline the development of structural analogues of Ramsey's theorem through to the current state of the art for colorings of the space of subcopies of a given infinite structure. Much of the work in this talk is joint with Andy Zucker, including our recent paper on infinite-dimensional structural Ramsey theory. An important result with Zucker we can weaken one of Todorcevic's four axioms and still guarantee a topological Ramsey space.
MC 5501