Some implications of Ramsey Choice for n-element sets

TL;DR

This paper explores the relationships between various weak choice principles for n-element sets, providing full characterizations and independence results within ZF set theory, and answering open questions on implications between these principles.

Contribution

It offers a complete characterization of when certain choice implications hold in ZF and proves new implications, answering open questions in the field.

Findings

Characterization of when RC_m implies WOC_n^- in ZF.

RC_5 implies LOC_5^- and RC_6 implies C_3^-.

RC_6 implies C_9^- and RC_7 implies LOC_7^-.

Abstract

Let $n\in\omega$. The weak choice principle $\operatorname{RC}_n$ states that for every infinite set $x$ there is an infinite subset $y\subseteq x$ with a choice function on $[y]^n:=\{z\subseteq y\mid \lvert z\rvert =n\}$. $\operatorname{C}_n^-$ states that for every infinite family of $n$-element sets, there is an infinite subfamily $\mathcal{G}\subseteq\mathcal{F}$ with a choice function. $\operatorname{LOC}_n^-$ and $\operatorname{WOC}_n^-$ are the same statement but we assume that the family $\mathcal{F}$ is linearly orderable ($\operatorname{LOC}_n^-$) or well-orderable ($\operatorname{WOC}_n^-$). In the first part of this paper we will give a full characterization of when the implication $\operatorname{RC}_m\Rightarrow \operatorname{WOC}_n^-$ with $m,n\in\omega$ holds in $\operatorname{ZF}$. We will prove the independence results by using suitable Fraenkel-Mostowski permutation models. In the second part of we will show some generalizations. In particular we will show that $\operatorname{RC}_5\Rightarrow \operatorname{LOC}_5^-$ and $\operatorname{RC}_6\Rightarrow \operatorname{C}_3^-$, answering two open questions from Halbeisen and Tachtsis. Furthermore we will show that $\operatorname{RC}_6\Rightarrow \operatorname{C}_9^-$ and $\operatorname{RC}_7\Rightarrow \operatorname{LOC}_7^-$.