When a set theorist hears “combinatorics”: Infinite Ramsey theory

Abstract

Ramsey theory studies how to find highly-ordered substructures within an otherwise unwieldy object. Ramsey theory is a highly active area of research in contemporary mathematics, with some mathematicians focusing on finite structures and others on infinite ones. In this survey paper, we will give an overview of a few topics in infinite Ramsey theory, with an emphasis on how set theory is involved. That is, we will focus on large, infinite objects and ask exactly how infinite they must be in order to ensure that we have infinite, highly-ordered substructures. After introducing the general idea in the finite case, we will prove Ramsey's theorem about infinite graphs. Then we will transition into questions about finding uncountably infinite, highly-ordered substructures. This will give us a convenient excuse to discuss infinities and independence results in set theory, as well as topological colorings. No knowledge of set theory or topology is required to understand this paper.