A combinatorial property of pκλ
A combinatorial property of pκλ
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pκλ 的组合特性
DOI：
10.1017/s0022481200051926
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发表时间：
1976
影响因子：
0.6
通讯作者：
T. Menas
中科院分区：
文献类型：
--
作者：
T. Menas
在一篇关于组合性质和大基数的论文[2]中，Jech将基数κ的几个组合性质推广到基数小于κ的λ的所有子集的集合的类似性质，记为“pκλ"，其中λ是任何基数≤κ。我们将在本文中考虑其中一个性质，该性质历史上植根于拉姆齐[10]的定理和Rowbottom [12]的工作中。如[2]中，定义[pκλ]2 = {{x，y}：x，y ∈ pκλ and x <$y}。pκλ的无界子集A对于函数F：[pκλ]2 → 2是齐次的，如果存在k < 2，使得对所有x，y ∈ A且x <$y或y <$x，F（{x，y}）= k。pκλ上的一个二值测度<$u是好的，如果它是κ-完备的，并且如果对所有α < λ，<$u（{x ∈ pκλ：α ∈ x}）= 1，并且<$u是正规的，如果另外，对于每个函数f：pκλ → λ使得<$u（{x ∈ p κ λ：f（x）∈ x}）= 1，存在一个α < λ使得<$u（{x ∈ pκλ：f（x）= α}）= 1。最后，pκλ上的精细测度具有分拆性质，如果每个F：[pκλ]2 → 2都有测度1的齐次集。
In a paper on combinatorial properties and large cardinals [2], Jech extended several combinatorial properties of a cardinal κ to analogous properties of the set of all subsets of λ of cardinality less than κ, denoted by “pκλ”, where λ is any cardinal ≤κ. We shall consider in this paper one of these properties which is historically rooted in a theorem of Ramsey [10] and in work of Rowbottom [12]. As in [2], define [pκλ]2 = {{x, y}: x, y ∈ pκλ and x ≠ y}. An unbounded subset A of pκλ is homogeneous for a function F: [pκλ]2 → 2 if there is a k < 2 so that for all x, y ∈ A with either x ⊊ y or y ⊊ x, F({x, y}) = k. A two-valued measure ü on pκλ is fine if it is κ-complete and if for all α < λ, ü({x ∈ pκλ: α ∈ x}) = 1, and ü is normal if, in addition, for every function f: pκλ → λsuch that ü({x ∈ pκλ: f(x) ∈ x}) = 1, there is an α < λ such that ü({x ∈ pκλ: f(x) = α}) = 1. Finally, a fine measure on pκλ has the partition property if every F: [pκλ]2 → 2 has a homogeneous set of measure one.