Introduction to big Ramsey degrees

We give an introduction to structural generalizations of the well known Ramsey theorem. We start by 1960’s work of Laver and Devlin about coloring finite subsets of rational numbers and show some recent results in the area. In particular a new and relatively straighforward proof of Dobrinen’s theorem stating that big Ramsey degrees of the triangle-free graphs are finite. We show generalizations of this proof to new Ramsey results and outline an emerging theory of big Ramsey structures.

This is a joint work with Balko, Chodounsky, Dobrinen, Konečný, Nešetřil, de Rancourt, Todorcevic and Zucker.