$\operatorname {nC}_{<\aleph _0}^-:$ 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$F\in \mathcal {G}$ .

$\operatorname {KWF}^-:$ 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$F\in \mathcal {G}$ .