In the area of mathematics known as

Ramsey theory Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics that focuses on the appearance of order in a substructure given a structure of a known size. Problems in Ramsey theory typically ask ...

, a Ramsey class is one which satisfies a generalization of

Ramsey's theorem In combinatorics, Ramsey's theorem, in one of its graph-theoretic forms, states that one will find monochromatic cliques in any edge labelling (with colours) of a sufficiently large complete graph. To demonstrate the theorem for two colours (s ...

. Suppose

A

B

and

C

are structures and

k

is a positive integer. We denote by

\binom

the set of all subobjects

A'

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

\binom

there exists a

B' \in \binom

and an

1 \leq i \leq k

such that

\binom \subseteq X_i

. Suppose

K

is a class of structures closed under

isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

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.

References

Ramsey theory {{combin-stub