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

Abstract

For nin omega , the weak choice principle textrm{RC}_n is defined as follows:For every infinite setXthere is an infinite subsetYsubseteq Xwith a choice function on[Y]^n:={zsubseteq Y:|z|=n}.The choice principle textrm{C}_n^- states the following:For every infinite family ofn-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,nin omega we will give a full characterization of when the implication textrm{RC}_mRightarrow 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}_5Rightarrow textrm{LOC}_5^- and that textrm{RC}_6Rightarrow 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}_6Rightarrow textrm{C}_9^- and that textrm{RC}_7Rightarrow textrm{LOC}_7^-.