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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Arrow notation (Ramsey theory) ： ウィキペディア英語版 Infinitary combinatorics In mathematics, infinitary combinatorics, or combinatorial set theory, is an extension of ideas in combinatorics to infinite sets. Some of the things studied include continuous graphs and trees, extensions of Ramsey's theorem, and Martin's axiom. Recent developments concern combinatorics of the continuum〔Andreas Blass, ''Combinatorial Cardinal Characteristics of the Continuum'', Chapter 6 in Handbook of Set Theory, edited by Matthew Foreman and Akihiro Kanamori, Springer, 2010〕 and combinatorics on successors of singular cardinals.〔Todd Eisworth, ''Successors of Singular Cardinals'' Chapter 15 in Handbook of Set Theory, edited by Matthew Foreman and Akihiro Kanamori, Springer, 2010〕 ==Ramsey theory for infinite sets== Write κ, λ for ordinals, ''m'' for a cardinal number and ''n'' for a natural number. introduced the notation :$\backslash kappa\backslash rightarrow(\backslash lambda)^n\_m$ as a shorthand way of saying that every partition of the set ()''n'' of ''n''-element subsets of $\backslash kappa$ into ''m'' pieces has a homogeneous set of order type λ. A homogeneous set is in this case a subset of κ such that every ''n''-element subset is in the same element of the partition. When ''m'' is 2 it is often omitted. Assuming the Axiom of Choice, there are no ordinals κ with κ→(ω)ω, so ''n'' is usually taken to be finite. An extension where ''n'' is almost allowed to be infinite is the notation :$\backslash kappa\backslash rightarrow(\backslash lambda)^\_m$ which is a shorthand way of saying that every partition of the set of finite subsets of κ into ''m'' pieces has a subset of order type λ such that for any finite ''n'', all subsets of size ''n'' are in the same element of the partition. When ''m'' is 2 it is often omitted. Another variation is the notation :$\backslash kappa\backslash rightarrow(\backslash lambda,\; \backslash mu)^n$ which is a shorthand way of saying that every coloring of the set ()''n'' of ''n''-element subsets of κ with 2 colors has a subset of order type λ such that all elements of ()''n'' have the first color, or a subset of order type μ such that all elements of ()''n'' have the second color. Some properties of this include: (in what follows $\backslash kappa$ is a cardinal) :$\backslash alef\_0\backslash rightarrow(\backslash alef\_0)^n\_k$ for all finite ''n'' and ''k'' (Ramsey's theorem). :$\backslash beth\_n^+\backslash rightarrow(\backslash alef\_1)\_^$ (Erdős–Rado theorem.) :$2^\backslash kappa\backslash not\backslash rightarrow(\backslash kappa^+)^2$ (Sierpiński theorem) :$2^\backslash kappa\backslash not\backslash rightarrow(3)^2\_\backslash kappa$ :$\backslash kappa\backslash rightarrow(\backslash kappa,\backslash alef\_0)^2$ (Erdős–Dushnik–Miller theorem). In choiceless universes, partition properties with infinite exponents may hold, and some of them are obtained as consequences of the Axiom of determinacy (AD). For example, Donald A. Martin proved that AD implies :$\backslash alef\_1\backslash rightarrow(\backslash alef\_1)^\_2$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Infinitary combinatorics」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.