Rosenthal families, pavings, and generic cardinal invariants,Proceedings of the American Mathematical Society

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Rosenthal families, pavings, and generic cardinal invariants
Proceedings of the American Mathematical Society ( IF 1 ) Pub Date : 2021-01-25 , DOI: 10.1090/proc/15252
Piotr Koszmider , Arturo Martínez-Celis

Abstract:Following D. Sobota we call a family $\mathcal F$ of infinite subsets of $\mathbb{N}$ a Rosenthal family if it can replace the family of all infinite subsets of $\mathbb{N}$ in the classical Rosenthal lemma concerning sequences of measures on pairwise disjoint sets. We resolve two problems on Rosenthal families: every ultrafilter is a Rosenthal family and the minimal size of a Rosenthal family is exactly equal to the reaping cardinal $\mathfrak{r}$ . This is achieved through analyzing nowhere reaping families of subsets of $\mathbb{N}$ and through applying a paving lemma which is a consequence of a paving lemma concerning linear operators on $\ell _1^n$ due to Bourgain. We use connections of the above results with free set results for functions on $\mathbb{N}$ and with linear operators on

to determine the values of several other derived cardinal invariants.

中文翻译：

罗森塔尔（Rosenthal）家族，铺路和通用基数不变式

摘要：在D. Sobota之后，如果它可以代替经典的Rosenthal引理中关于成对不相交集的度量序列的所有无限子集的族，我们就称其为Rosenthal族的无限子集。我们解决了有关Rosenthal系列的两个问题：每个超滤器都是一个Rosenthal系列，并且Rosenthal系列的最小尺寸正好等于收割的基数。这是通过无处可收的子集族分析和通过应用铺装引理实现的，这是由于布尔加因引起的涉及线性算子的铺装引理的结果。我们将以上结果与自由设置结果的连接用于函数和上的线性运算符 $\数学F$ $\ mathbb {N}$ $\ mathbb {N}$ $\ mathfrak {r}$ $\ mathbb {N}$ $\ ell _1 ^ n$ $\ mathbb {N}$

确定其他几个衍生的基不变量的值。

更新日期：2021-02-08

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