Algebra Seminar talk
Dragan Mašulović (Univerzitet u Novom Sadu)
Categorical Ramsey Theory
Generalizing the classical results of F.P. Ramsey from the late 1920's, the structural Ramsey theory originated at the beginning of 1970’s. We say that a class $K$ of finite structures has the Ramsey property if the following holds: for any number $k \ge 2$ of colors and all $A, B \in K$ such that $A$ embeds into $B$ there is a $C \in K$ such that no matter how we color the copies of $A$ in $C$ with $k$ colors, there is a monochromatic copy $B'$ of $B$ in $C$ (that is, all the copies of $A$ that fall within $B'$ are colored by the same color).
Showing that the Ramsey property holds for a class of finite structures $K$ can be an extremely challenging task and a slew of sophisticated methods have been proposed in literature. These methods are usually constructive: given $A, B \in K$ and $k \ge 2$ they prove the Ramsey property directly by constructing a structure $C \in K$ with the desired properties.
It was Leeb who pointed out already in early 1970's that the use of category theory can be quite helpful both in the formulation and in the proofs of results pertaining to structural Ramsey theory. Instead of pursuing the original approach by Leeb (which has very fruitfully been applied to a wide range of Ramsey problems) we proposed in the last few years a set of new strategies to show that a class of structures has the Ramsey property.
In this talk we explicitly put the Ramsey property and the dual Ramsey property in the context of categories of finite structures. We use elementary category theory to generalize some combinatorial results and using the machinery of very basic category theory provide new combinatorial statements (whose formulations do not refer to category-theoretic notions) concerning both the Ramsey property and the dual Ramsey property.