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Milliken–Taylor theorem ：ウィキペディア英語版

Milliken–Taylor theorem

In mathematics, the Milliken–Taylor theorem in combinatorics is a generalization of both Ramsey's theorem and Hindman's theorem. It is named after Keith Milliken and Alan D. Taylor.
Let

\mathcal_f(\mathbb)

denote the set of finite subsets of

\mathbb

, and define a partial order on

\mathcal_f(\mathbb)

by α<β if and only if max α\langle a_n \rangle_^\infty \subset \mathbb and , let
:

)^k_< = \left \_f(\mathbb)\text\alpha_1 <\cdots < \alpha_k \right \}.

Let

()^k

denote the ''k''-element subsets of a set ''S''. The Milliken–Taylor theorem says that for any finite partition

)^k=C_1 \cup C_2 \cup \cdots \cup C_r

, there exist some and a sequence

\langle a_n \rangle_^ \subset \mathbb

such that

)^k_< \subset C_i

.
For each

\langle a_n \rangle_^\infty \subset \mathbb

, call

(a_n \rangle_^\infty))^k_<

an ''MT^k set''. Then, alternatively, the Milliken–Taylor theorem asserts that the collection of MT^''k'' sets is partition regular for each ''k''.
==References==

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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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