A New Weak Choice Principle

TL;DR

This paper introduces a new weak choice principle $ ext{nRC}_{fin}$, explores its relations to existing principles through permutation models, and establishes stronger analogues of known results in set theory.

Contribution

It defines the new principle $ ext{nRC}_{fin}$ and analyzes its relationship with $ ext{RC}_m$ and $ ext{kC}_{fin}^-$ using permutation models, extending previous work.

Findings

$ ext{nRC}_{fin}$ is consistent with ZF and independent of some weak choice principles.

Established new relations between $ ext{nRC}_{fin}$ and $ ext{RC}_m$ principles.

Proved stronger analogues of Montenegro's results for $ ext{kC}_{fin}^-$.

Abstract

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 $A\in\mathcal{A}$.