Ramsey class について

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Ramsey class ：ウィキペディア英語版

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

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 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.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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