Ramsey class

 ( Ramsey Theory ) 

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 $\backslash binom\{B\}\{A\}$ the set of all subobjects $A\text{'}$ of $B$ which are isomorphic to $A$. We further denote by $C\; \backslash rightarrow\; (B)^A\_k$ the property that for all partitions $X\_1\; \backslash cup\; X\_2\backslash cup\; \backslash dots\backslash cup\; X\_k$ of $\backslash binom\{C\}\{A\}$ there exists a $B\text{'}\; \backslash in\; \backslash binom\{C\}\{B\}$ and an $1\; \backslash leq\; i\; \backslash leq\; k$ such that $\backslash binom\{B\text{'}\}\{A\}\; \backslash subseteq\; X\_i$.

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 $B\backslash in\; K$ there is a $C\; \backslash in\; K$ such that $C\; \backslash rightarrow\; (B)^A\_k$ holds. If $K$ has the $A$-Ramsey property for all $A\; \backslash in\; K$ 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.