A NEW WEAK CHOICE PRINCIPLE

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