Some implications of Ramsey Choice for n-element sets

Abstract

Let $n\in \omega$. The weak choice principle ${\mathrm{RC}}_n$ states that for every infinite set $x$ there is an infinite subset $y\subseteq x$ with a choice function on $[y{]}^n:=\{z\subseteq y\mid |z|=n\}$. $C_n^{-}$ states that for every infinite family of $n$-element sets, there is an infinite subfamily $\mathcal{G}\subseteq \mathcal{F}$ with a choice function. ${\mathrm{LOC}}_n^{-}$ and ${\mathrm{WOC}}_n^{-}$ are the same statement but we assume that the family $\mathcal{F}$ is linearly orderable (${\mathrm{LOC}}_n^{-}$) or well-orderable (${\mathrm{WOC}}_n^{-}$). In the first part of this paper we will give a full characterization of when the implication ${\mathrm{RC}}_m\Rightarrow {\mathrm{WOC}}_n^{-}$ with $m,n\in \omega$ holds in $\mathrm{ZF}$. 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 ${\mathrm{RC}}_5\Rightarrow {\mathrm{LOC}}_5^{-}$ and ${\mathrm{RC}}_6\Rightarrow C_3^{-}$, answering two open questions from Halbeisen and Tachtsis. Furthermore we will show that ${\mathrm{RC}}_6\Rightarrow C_9^{-}$ and ${\mathrm{RC}}_7\Rightarrow {\mathrm{LOC}}_7^{-}$.