In the area of mathematics known as Ramsey theory, a Ramsey class is one which satisfies a generalization of Ramsey's theorem. Suppose A {\displaystyle A} , B {\displaystyle B} and C {\displaystyle C} are structures and k {\displaystyle k} is a positive integer. We denote by ( B A ) {\displaystyle {\binom {B}{A}}} the set of all subobjects A ′ {\displaystyle A'} of B {\displaystyle B} which are isomorphic to A {\displaystyle A} . We further denote by C → ( B ) k A {\displaystyle C\rightarrow (B)_{k}^{A}} the property that for all partitions X 1 ∪ X 2 ∪ ⋯ ∪ X k {\displaystyle X_{1}\cup X_{2}\cup \dots \cup X_{k}} of ( C A ) {\displaystyle {\binom {C}{A}}} there exists a B ′ ∈ ( C B ) {\displaystyle B'\in {\binom {C}{B}}} and an 1 ≤ i ≤ k {\displaystyle 1\leq i\leq k} such that ( B ′ A ) ⊆ X i {\displaystyle {\binom {B'}{A}}\subseteq X_{i}} . Suppose K {\displaystyle K} is a class of structures closed under isomorphism and substructures. We say the class K {\displaystyle K} has the A-Ramsey property if for ever positive integer k {\displaystyle k} and for every B ∈ K {\displaystyle B\in K} there is a C ∈ K {\displaystyle C\in K} such that C → ( B ) k A {\displaystyle C\rightarrow (B)_{k}^{A}} holds. If K {\displaystyle K} has the A {\displaystyle A} -Ramsey property for all A ∈ K {\displaystyle A\in K} then we say K {\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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