  Let $n\in\omega$. The weak choice principle $\operatorname{RC}_n$ states that
for every infinite set $x$ there is an infinite subset $y\subseteq x$ with a
choice function on $[y]^n:=\{z\subseteq y\mid \lvert z\rvert =n\}$.
$\operatorname{C}_n^-$ states that for every infinite family of $n$-element
sets, there is an infinite subfamily $\mathcal{G}\subseteq\mathcal{F}$ with a
choice function. $\operatorname{LOC}_n^-$ and $\operatorname{WOC}_n^-$ are the
same statement but we assume that the family $\mathcal{F}$ is linearly
orderable ($\operatorname{LOC}_n^-$) or well-orderable
($\operatorname{WOC}_n^-$).
  In the first part of this paper we will give a full characterization of when
the implication $\operatorname{RC}_m\Rightarrow \operatorname{WOC}_n^-$ with
$m,n\in\omega$ holds in $\operatorname{ZF}$. We will prove the independence
results by using suitable Fraenkel-Mostowski permutation models. In the second
part of we will show some generalizations. In particular we will show that
$\operatorname{RC}_5\Rightarrow \operatorname{LOC}_5^-$ and
$\operatorname{RC}_6\Rightarrow \operatorname{C}_3^-$, answering two open
questions from Halbeisen and Tachtsis. Furthermore we will show that
$\operatorname{RC}_6\Rightarrow \operatorname{C}_9^-$ and
$\operatorname{RC}_7\Rightarrow \operatorname{LOC}_7^-$.
