# Preservation of Partition Properties under Products

We study classes of finite structures in a relational language that are closed under isomorphism.

We will present some well-known product operations on these classes, sometimes with slight modifications. We will then show that certain properties from structural partition theory are preserved under some of these products and not others. One such property is age indivisibility: given a class $\mathbf{K}$ as above, we say that $\mathbf{K}$ is *age indivisible* if, for all $A \in \mathbf{K}$ and $k \ge 2$, there exists $B \in \mathbf{K}$ such that, for all $c : B \rightarrow k$, there exists $f \in \textrm{emb}(A,B)$ such that $|c(f(A))| = 1$. We will also give some motivation for this investigation in the form of questions about $\mathbf{K}$-configurations, which can be used to express certain dividing lines between first-order theories.

This is joint work with Vincent Guingona and Miriam Parnes.