  For every natural number $n$ we introduce a new weak choice principle
$\mathrm{nRC_{fin}}$: Given any infinite set $x$, there is an infinite subset
$y\subseteq x$ and a selection function $f$ that chooses an $n$-element subset
from every finite $z\subseteq y$ containing at least $n$ elements. By
constructing new permutation models built on a set of atoms obtained as
Fra\"iss\'e limits, we will study the relation of $\mathrm{nRC_{fin}}$ to the
weak choice principles $\mathrm{RC_m}$ (that has already been studied by
Montenegro, Halbeisen and Tachtsis): Given any infinite set $x$, there is an
infinite subset $y\subseteq x$ with a choice function $f$ on the family of all
$m$-element subsets of $y$. Moreover, we prove a stronger analogue of
Montenegros results when we study the relation between $\mathrm{nRC_{fin}}$ and
$\mathrm{kC_{fin}^-}$ which is defined by: Given any infinite family
$\mathcal{F}$ of finite sets of cardinality greater than $k$, there is an
infinite subfamily $\mathcal{A}\subseteq \mathcal{F}$ with a selection function
$f$ that chooses a $k$-element subset from each $A\in\mathcal{A}$.
