Abstract For every natural number n we introduce a new weak choice principle n upper R upper C Subscript f i n $\mathrm {nRC_{fin}}$ n R C f i n : Given any infinite set x , there is an infinite subset y subset of or equal to x $y\subseteq x$ y ⊆ x and a selection function f that chooses an n-element subset from every finite z subset of or equal to y $z\subseteq y$ z ⊆ y containing at least n elements. By constructing new permutation models built on a set of atoms obtained as Fraïssé limits, we will study the relation of n upper R upper C Subscript f i n $\mathrm {nRC_{fin}}$ n R C f i n to the weak choice principles upper R upper C Subscript m $\mathrm {RC_m}$ R C m (that has already been studied in [3] and [6]): Given any infinite set x , there is an infinite subset y subset of or equal to x $y\subseteq x$ y ⊆ x with a choice function f on the family of all m-element subsets of y . Moreover, we prove a stronger analogue of the results in [6] when we study the relation between

For a set $M$, $\operatorname{fin}(M)$ denotes the set of all finite subsets of $M$, $M^2$ denotes the Cartesian product $M\times M$, $[M]^2$ denotes the set of all $2$-element subsets of $M$, and $\operatorname{seq}^{1-1}(M)$ denotes the set of all finite sequences without repetition which can be formed with elements of $M$. Furthermore, for a set $S$, let $|S|$ denote the cardinality of $S$. Under the assumption that the four cardinalities $|[M]^2|$, $|M^2|$, $|\operatorname{fin}(M)|$, $|\operatorname{seq}^{1-1}(M)|$ are pairwise distinct and pairwise comparable in ZF, there are six possible linear orderings between these four cardinalities. We show that at least five of the six possible linear orderings are consistent with ZF.

Abstract 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.