Logic Seminar

Dragan MasulovicUniversity of Novi Sad
Structural Ramsey Theory from the point of view of Category Theory

Thursday, May 12, 2022 - 2:45pm
Malott 206

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 \emph{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 \emph{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 $C \in K$ which is Ramsey for $B$, $A$ and $k$.

In this talk we explicitly put the Ramsey property and the dual Ramsey property in the context of categories of finite structures. We use the machinery of category theory to provide new Ramsey and dual Ramsey statements for some classes of finite algebras and related combinatorial structures.