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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 Partition regularity ： ウィキペディア英語版 Partition regularity In combinatorics, a branch of mathematics, partition regularity is one notion of largeness for a collection of sets. Given a set $X$, a collection of subsets $\backslash mathbb\; \backslash subset\; \backslash mathcal(X)$ is called ''partition regular'' if every set ''A'' in the collection has the property that, no matter how ''A'' is partitioned into finitely many subsets, at least one of the subsets will also belong to the collection. That is, for any $A\; \backslash in\; \backslash mathbb$, and any finite partition $A\; =\; C\_1\; \backslash cup\; C\_2\; \backslash cup\; \backslash cdots\; \backslash cup\; C\_n$, there exists an ''i'' ≤ ''n'', such that $C\_i$ belongs to $\backslash mathbb$. Ramsey theory is sometimes characterized as the study of which collections $\backslash mathbb$ are partition regular. == Examples == * the collection of all infinite subsets of an infinite set ''X'' is a prototypical example. In this case partition regularity asserts that every finite partition of an infinite set has an infinite cell (i.e. the infinite pigeonhole principle.) * sets with positive upper density in $\backslash mathbb$: the ''upper density'' $\backslash overline(A)$ of $A\; \backslash subset\; \backslash mathbb$ is defined as $\backslash overline(A)\; =\; \backslash limsup\_\; \backslash frac.$ * For any ultrafilter $\backslash mathbb$ on a set $X$, $\backslash mathbb$ is partition regular. If $\backslash mathbb\; \backslash ni\; A\; =\backslash bigcup\_1^n\; C\_i$, then for exactly one $i$ is $C\_i\; \backslash in\; \backslash mathbb$. * sets of recurrence: a set R of integers is called a ''set of recurrence'' if for any measure preserving transformation $T$ of the probability space (Ω, β, μ) and $A\; \backslash in\backslash \; \backslash beta$ of positive measure there is a nonzero $n\; \backslash in\; R$ so that $\backslash mu(A\; \backslash cap\; T^A)\; >\; 0$. * Call a subset of natural numbers ''a.p.-rich'' if it contains arbitrarily long arithmetic progressions. Then the collection of a.p.-rich subsets is partition regular (Van der Waerden, 1927). * Let $()^n$ be the set of all ''n''-subsets of $A\; \backslash subset\; \backslash mathbb$. Let $\backslash mathbb^n\; =\; \backslash bigcup^\_^n$ is partition regular. (Ramsey, 1930). * For each infinite cardinal $\backslash kappa$, the collection of stationary sets of $\backslash kappa$ is partition regular. More is true: if $S$ is stationary and $S=\backslash bigcup\_\; S\_$ for some , then some $S\_$ is stationary. * the collection of $\backslash Delta$-sets: $A\; \backslash subset\; \backslash mathbb$ is a $\backslash Delta$-set if $A$ contains the set of differences $\backslash$ for some sequence $\backslash langle\; s\_n\; \backslash rangle^\backslash omega\_$. * the set of barriers on $\backslash mathbb$: call a collection $\backslash mathbb$ of finite subsets of $\backslash mathbb$ a ''barrier'' if: * * $\backslash forall\; X,Y\; \backslash in\; \backslash mathbb,\; X\; \backslash not\backslash subset\; Y$ and * * for all infinite $I\; \backslash subset\; \backslash cup\; \backslash mathbb$, there is some $X\; \backslash in\; \backslash mathbb$ such that the elements of X are the smallest elements of I; ''i.e.'' $X\; \backslash subset\; I$ and : This generalizes Ramsey's theorem, as each $()^n$ is a barrier. (Nash-Williams, 1965) * finite products of infinite trees (Halpern–Läuchli, 1966) * piecewise syndetic sets (Brown, 1968) * Call a subset of natural numbers ''i.p.-rich'' if it contains arbitrarily large finite sets together with all their finite sums. Then the collection of i.p.-rich subsets is partition regular (Folkman–Rado–Sanders, 1968). * (''m'', ''p'', ''c'')-sets (Deuber, 1973) * IP sets (Hindman, 1974, see also Hindman, Strauss, 1998) * MT''k'' sets for each ''k'', ''i.e.'' ''k''-tuples of finite sums (Milliken–Taylor, 1975) * central sets; ''i.e.'' the members of any minimal idempotent in $\backslash beta\backslash mathbb$, the Stone–Čech compactification of the integers. (Furstenberg, 1981, see also Hindman, Strauss, 1998) 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「Partition regularity」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.