This is a talk in the Boise State Topics in Algebra, Topology, Etc. Research Seminar on 2023 November 17.
In 1966, Abraham Robinson used ideas from model theory to come up with nonstandard analysis, an approach to analysis allowing infinitesimals as actually existing objects. This talk is not about that. Instead, this talk is about a different area where nonstandard methods have been fruitful, namely combinatorics. With Timothy Trujillo, we were interested in whether nonstandard methods could be applied to understand work in topological Ramsey theory. After all, this area studies a generalization of the combinatorial-topological Ellentuck space on the integers, and nonstandard methods have enjoyed use in integer combinatorics.
In this talk I will give an introduction to the use of nonstandard methods and how they can be used to prove results like Ramsey’s theorem. I’ll discuss how these ideas can be used to prove the Nash-Williams separation theorem, and I’ll gesture toward how to generalize this to the setting of abstract Ramsey spaces.