Some implications of Ramsey Choice for families of \(\varvec{n}\)-element sets

Abstract

For \(n\in \omega \), the weak choice principle \(\textrm{RC}_n\) is defined as follows:

For every infinite set X there is an infinite subset \(Y\subseteq X\) with a choice function on \([Y]^n:=\{z\subseteq Y:|z|=n\}\).

The choice principle \(\textrm{C}_n^-\) states the following:

For every infinite family of n-element sets, there is an infinite subfamily \({\mathcal {G}}\subseteq {\mathcal {F}}\) with a choice function.

The choice principles \(\textrm{LOC}_n^-\) and \(\textrm{WOC}_n^-\) are the same as \(\textrm{C}_n^-\), but we assume that the family \({\mathcal {F}}\) is linearly orderable (for \(\textrm{LOC}_n^-\)) or well-orderable (for \(\textrm{WOC}_n^-\)). In the first part of this paper, for \(m,n\in \omega \) we will give a full characterization of when the implication \(\textrm{RC}_m\Rightarrow \textrm{WOC}_n^-\) holds in \({\textsf {ZF}}\). We will prove the independence results by using suitable Fraenkel-Mostowski permutation models. In the second part, we will show some generalizations. In particular, we will show that \(\textrm{RC}_5\Rightarrow \textrm{LOC}_5^-\) and that \(\textrm{RC}_6\Rightarrow \textrm{C}_3^-\), answering two open questions from Halbeisen and Tachtsis (Arch Math Logik 59(5):583–606, 2020). Furthermore, we will show that \(\textrm{RC}_6\Rightarrow \textrm{C}_9^-\) and that \(\textrm{RC}_7\Rightarrow \textrm{LOC}_7^-\).