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 $$C \rightarrow (B)^A_k$$ 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 $$C \rightarrow (B)^A_k$$ 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.