For \(n\in \omega \), the weak choice principle \(\textrm{RC}_n\) is defined as follows:

For every infinite set X there is an infinite subset \(Y\subseteq X\) with a choice function on \([Y]^n:=\{z\subseteq Y:|z|=n\}\).

The choice principle \(\textrm{C}_n^-\) states the following:

For every infinite family of n-element sets, there is an infinite subfamily \({\mathcal {G}}\subseteq {\mathcal {F}}\) with a choice function.

The choice principles \(\textrm{LOC}_n^-\) and \(\textrm{WOC}_n^-\) are the same as \(\textrm{C}_n^-\), but we assume that the family \({\mathcal {F}}\) is linearly orderable (for \(\textrm{LOC}_n^-\)) or well-orderable (for \(\textrm{WOC}_n^-\)). In the first part of this paper, for \(m,n\in \omega \) we will give a full characterization of when the implication \(\textrm{RC}_m\Rightarrow \textrm{WOC}_n^-\) holds in \({\textsf {ZF}}\). We will prove the independence results by using suitable Fraenkel-Mostowski permutation models. In the second part, we will show some generalizations. In particular, we will show that \(\textrm{RC}_5\Rightarrow \textrm{LOC}_5^-\) and that \(\textrm{RC}_6\Rightarrow \textrm{C}_3^-\), answering two open questions from Halbeisen and Tachtsis (Arch Math Logik 59(5):583–606, 2020). Furthermore, we will show that \(\textrm{RC}_6\Rightarrow \textrm{C}_9^-\) and that \(\textrm{RC}_7\Rightarrow \textrm{LOC}_7^-\).

对于\(n\in \omega \)，弱选择原则\(\textrm{RC}_n\)定义如下：

对于每个无限集 X 都有一个无限子集 \(Y\subseteq X\) ，其选择函数在 \([Y]^n:=\{z\subseteq Y:|z|=n\}\)上。

选择原则\(\textrm{C}_n^-\)陈述如下：

对于 n 个元素集的每个无限族，都有一个具有选择函数的无限子族 \({\mathcal {G}}\subseteq {\mathcal {F}}\) 。

选择原则\(\textrm{LOC}_n^-\)和\(\textrm{WOC}_n^-\)与\(\textrm{C}_n^-\)相同，但我们假设族\({\mathcal {F}}\)是线性可排序的（对于\(\textrm{LOC}_n^-\)）或可良好排序的（对于\(\textrm{WOC}_n^-\)）。在本文的第一部分，对于\(m,n\in \omega \)我们将给出一个完整的表征，即当蕴涵\(\textrm{RC}_m\Rightarrow \textrm{WOC}_n^-\)持有\({\textsf {ZF}}\). 我们将通过使用合适的 Fraenkel-Mostowski 置换模型来证明独立性结果。在第二部分，我们将展示一些概括。特别是，我们将展示\(\textrm{RC}_5\Rightarrow \textrm{LOC}_5^-\)和\(\textrm{RC}_6\Rightarrow \textrm{C}_3^-\)，回答 Halbeisen 和 Tachtsis 提出的两个开放性问题 (Arch Math Logik 59(5):583–606, 2020)。此外，我们将展示\(\textrm{RC}_6\Rightarrow \textrm{C}_9^-\)和\(\textrm{RC}_7\Rightarrow \textrm{LOC}_7^-\)。