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 $(\genfrac{}{}{0ex}{}{B}{A})$ the set of all subobjects $A^{\prime }$ of $B$ which are isomorphic to $A$. We further denote by $C\to (B{)}_k^A$ the property that for all partitions $X_1\cup X_2\cup \dots \cup X_k$ of $(\genfrac{}{}{0ex}{}{C}{A})$ there exists a $B^{\prime }\in (\genfrac{}{}{0ex}{}{C}{B})$ and an $1\le i\le k$ such that $(\genfrac{}{}{0ex}{}{B^{\prime }}{A})\subseteq X_i$.

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 $C\to (B{)}_k^A$ 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.