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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 \( \binom{B}{A} \) the set of all subobjects A' of B which are isomorphic to A. We further denote by \( C \rightarrow (B)^A_k \) the property that for all partitions \(X_1 \cup X_2\cup \dots\cup X_k \) of \( \binom{C}{A} \) there exists \( a B' \in \binom{C}{B} \) and an \(1 \leq i \leq k \) such that \(\binom{B'}{A} \subseteq X_i \).
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 \(C \rightarrow (B)^A_k \) 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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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