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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 Milliken's tree theorem ： ウィキペディア英語版 Milliken's tree theorem In mathematics, Milliken's tree theorem in combinatorics is a partition theorem generalizing Ramsey's theorem to infinite trees, objects with more structure than sets. Let T be a finitely splitting rooted tree of height ω, n a positive integer, and $\backslash mathbb^n\_T$ the collection of all strongly embedded subtrees of T of height n. In one of its simple forms, Milliken's tree theorem states that if $\backslash mathbb^n\_T=C\_1\; \backslash cup\; ...\; \backslash cup\; C\_r$ then for some strongly embedded infinite subtree R of T, $\backslash mathbb^n\_R\; \backslash subset\; C\_i$ for some i ≤ r. This immediately implies Ramsey's theorem; take the tree T to be a linear ordering on ω vertices. Define $\backslash mathbb^n=\; \backslash bigcup\_T\; \backslash mathbb^n\_T$ where T ranges over finitely splitting rooted trees of height ω. Milliken's tree theorem says that not only is $\backslash mathbb^n$ partition regular for each n < ω, but that the homogeneous subtree R guaranteed by the theorem is strongly embedded in T. == Strong embedding == Call T an α-tree if each branch of T has cardinality α. Define Succ(p, P)= $\backslash$, and $IS(p,P)$ to be the set of immediate successors of p in P. Suppose S is an α-tree and T is a β-tree, with 0 ≤ α ≤ β ≤ ω. S is ''strongly embedded'' in T if: * $S\; \backslash subset\; T$, and the partial order on S is induced from T, * if $s\; \backslash in\; S$ is nonmaximal in S and $t\; \backslash in\; IS(s,T)$, then $|Succ(t,T)\; \backslash cap\; IS(s,S)|=1$, * there exists a strictly increasing function from $\backslash alpha$ to $\backslash beta$, such that $S(n)\; \backslash subset\; T(f(n)).$ Intuitively, for S to be strongly embedded in T, * S must be a subset of T with the induced partial order * S must preserve the branching structure of T; ''i.e.'', if a nonmaximal node in S has n immediate successors in T, then it has n immediate successors in S * S preserves the level structure of T; all nodes on a common level of S must be on a common level in T. 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「Milliken's tree theorem」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.