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 ( B A ) the set of all subobjects A ′ of B which are isomorphic to A . We further denote by C → ( B ) k A the property that for all partitions X 1 ∪ X 2 ∪ ⋯ ∪ X k of ( C A ) there exists a B ′ ∈ ( C B ) and an 1 ≤ i ≤ k such that ( B ′ A ) ⊆ 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 ∈ K there is a C ∈ K such that C → ( B ) k A holds. If K has the A -Ramsey property for all A ∈ 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.
