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 formula_1, formula_2 and formula_3 are structures and formula_4 is a positive integer. We denote by formula_5 the set of all subobjects formula_6 of formula_2 which are isomorphic to formula_1. We further denote by formula_9 the property that for all partitions formula_10 of formula_11 there exists a formula_12 and an formula_13 such that formula_14. Suppose formula_15 is a class of structures closed under isomorphism and substructures. We say the class formula_15 has the A-Ramsey property  if for ever positive integer formula_4 and for every formula_18 there is a formula_19 such that formula_9 holds. If formula_15 has the formula_1-Ramsey property for all formula_23 then we say formula_15 is a Ramsey class. Ramsey's theorem is equivalent to the statement that the class of all finite sets is a Ramsey class.