A New Weak Choice Principle

Abstract

For every natural number $n$ we introduce a new weak choice principle $\mathrm{nR}{\mathrm{C}}_{\mathrm{fin}}$: 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 Fraïssé limits, we will study the relation of $\mathrm{nR}{\mathrm{C}}_{\mathrm{fin}}$ to the weak choice principles $\mathrm{R}{\mathrm{C}}_{\mathrm{m}}$ (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 $\mathrm{nR}{\mathrm{C}}_{\mathrm{fin}}$ and $\mathrm{k}{\mathrm{C}}_{\mathrm{fin}}^{-}$ 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 $A\in \mathcal{A}$.