For every $n\in \omega \setminus \{0,1\}$ we introduce the following weak choice principle:$\operatorname {nC}_{<\aleph _0}^-:$ For every infinite family$\mathcal {F}$ of finite sets of size at least n there is an infinite subfamily$\mathcal {G}\subseteq \mathcal {F}$ with a selection function$f:\mathcal {G}\to \left [\bigcup \mathcal {G}\right ]^n$ such that$f(F)\in [F]^n$ for all$F\in \mathcal {G}$ .Moreover, we consider the following choice principle:$\operatorname {KWF}^-:$ For every infinite family$\mathcal {F}$ of finite sets of size at least$2$ there is an infinite subfamily$\mathcal {G}\subseteq \mathcal {F}$ with a Kinna–Wagner selection function. That is, there is a function$g\colon \mathcal {G}\to \mathcal {P}\left (\bigcup \mathcal {G}\right )$ with$\emptyset \not =f(F)\subsetneq F$ for every$F\in \mathcal {G}$ .We will discuss the relations between these two choice principles and their relations to other well-known weak choice principles. Moreover, we will discuss what happens when we replace $\mathcal {F}$ by a linearly ordered or a well-ordered family.

中文翻译：

---

两个递减选择原则之间的关系

对于每一个$n\in \omega \setminus \{0,1\}$我们引入以下弱选择原则：$\operatorname {nC}_{<\aleph _0}^-:$对于每一个无限的家庭$\数学{F}$大小至少为 n 的有限集有一个无限的子族$\mathcal {G}\subseteq \mathcal {F}$带选择功能$f:\mathcal {G}\to \left [\bigcup \mathcal {G}\right ]^n$这样$f(F)\in [F]^n$对所有人$F\in \mathcal {G}$.此外，我们考虑以下选择原则：$\运营商名称 {KWF}^-:$对于每一个无限的家庭$\数学{F}$至少具有有限的大小集$2$有一个无限的亚科$\mathcal {G}\subseteq \mathcal {F}$具有 Kinna–Wagner 选择函数。也就是说，有一个函数$g\colon \mathcal {G}\to \mathcal {P}\left (\bigcup \mathcal {G}\right )$和$\emptyset \not =f(F)\subsetneq F$对于每个$F\in \mathcal {G}$.我们将讨论这两个选择原则之间的关系以及它们与其他众所周知的弱选择原则的关系。此外，我们将讨论替换时会发生什么$\数学{F}$由线性有序或良序族。