We discuss a new method used to prove that big Ramsey degrees of a given structure are finite. We start with a simple new proof of the theorem by Dobrinen showing the big Ramsey degrees of the homogeneous triangle free graphs are finite. This is based on an application of the Carlson-Simpson theorem. We outline how this proof can be carried to other structures including partial orders and metric spaces. Then we discuss a new theorem for trees with a successor operation that can be used to give bounds on big Ramsey degrees for structures with bigger forbidden configurations and in languages with higher arity.
Students at Uni Wien are required to attend in person.