Ramsey class

Short description: Class satisfying a generalization of Ramsey's theorem

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

Suppose [math]\displaystyle{ A }[/math], [math]\displaystyle{ B }[/math] and [math]\displaystyle{ C }[/math] are structures and [math]\displaystyle{ k }[/math] is a positive integer. We denote by [math]\displaystyle{ \binom{B}{A} }[/math] the set of all subobjects [math]\displaystyle{ A' }[/math] of [math]\displaystyle{ B }[/math] which are isomorphic to [math]\displaystyle{ A }[/math]. We further denote by [math]\displaystyle{ C \rightarrow (B)^A_k }[/math] the property that for all partitions [math]\displaystyle{ X_1 \cup X_2\cup \dots\cup X_k }[/math] of [math]\displaystyle{ \binom{C}{A} }[/math] there exists a [math]\displaystyle{ B' \in \binom{C}{B} }[/math] and an [math]\displaystyle{ 1 \leq i \leq k }[/math] such that [math]\displaystyle{ \binom{B'}{A} \subseteq X_i }[/math].

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

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