Ramsey class

In the area of mathematics known as Ramsey theory, a Ramsey class is one which satisfies a generalization of Ramsey's theorem.

Suppose $A$, $B$ and $C$ are structures and $k$ is a positive integer. We denote by ${\binom {B}{A}}$ the set of all subobjects $A'$ of $B$ which are isomorphic to $A$. We further denote by $C\rightarrow (B)_{k}^{A}$ the property that for all partitions $X_{1}\cup X_{2}\cup \dots \cup X_{k}$ of ${\binom {C}{A}}$ there exists a $B'\in {\binom {C}{B}}$ and an $1\leq i\leq k$ such that ${\binom {B'}{A}}\subseteq X_{i}$.

Suppose $K$ is a class of structures closed under isomorphism and substructures. We say the class $K$ has the A-Ramsey property if for ever positive integer $k$ and for every $B\in K$ there is a $C\in K$ such that $C\rightarrow (B)_{k}^{A}$ holds. If $K$ has the $A$-Ramsey property for all $A\in K$ then we say $K$ is a Ramsey class.

Ramsey's theorem is equivalent to the statement that the class of all finite sets is a Ramsey class.