Ramsey class

In the area of mathematics known as Ramsey theory, a Ramsey class^[1] is one which satisfies a generalization of Ramsey's theorem.

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

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

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

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