Tutte Colloquium - Andy Zucker

Title: Ramsey degrees, big and small

Abstract: Many of the seminal results in finite Ramsey theory can be phrased by saying that a certain class of finite structures has the Ramsey property, such as the ordinary finite Ramsey theorem (the class of finite linear orders), the dual Ramsey theorem (the class of finite lex-ordered Boolean algebras), the Graham-Leeb-Rothschild theorem (the class of lex-ordered, finite-dimensional vector spaces over a fixed finite field), and the Nesetril-Rodl theorem (the class of finite ordered triangle-free graphs, among many others). Furthermore, in many cases where classes of finite structures fail to have the Ramsey property, a quantitative weakening holds known as having finite small Ramsey degrees, where instead of looking for large monochromatic configurations in a given coloring, we look for large configurations in a small number of colors. For certain classes of finite structures (for instance, all of the classes mentioned above), the corresponding finite structures can be assembled into a canonical countably infinite structure called the Fraisse limit of the class. Then it becomes interesting and highly non-trivial to ask for which Fraisse structures do analogs of the ordinary infinite Ramsey theorem hold? Similarly, we cannot always insist on getting down to one color, so one asks for a characterization of the big Ramsey degree of every object in the given class. In recent joint work with Balko, Chodounsky, Dobrinen, Hubicka, Konecny, and Vena, we give an exact answer to this question for finitely constrained binary free amalgamation classes, thus yielding an "infinitary Nesetril-Rodl theorem."