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

Abstract

AbstractFor $$n\in \omega $$ n ∈ ω , the weak choice principle $$\textrm{RC}_n$$ RC n is defined as follows: For every infinite setXthere is an infinite subset$$Y\subseteq X$$ Y ⊆ X with a choice function on$$[Y]^n:=\{z\subseteq Y:|z|=n\}$$ [ Y ] n : = { z ⊆ Y : | z | = n } . The choice principle $$\textrm{C}_n^-$$ C n - states the following: For every infinite family ofn-element sets, there is an infinite subfamily$${\mathcal {G}}\subseteq {\mathcal {F}}$$ G ⊆ F with a choice function. The choice principles $$\textrm{LOC}_n^-$$ LOC n - and $$\textrm{WOC}_n^-$$ WOC n - are the same as $$\textrm{C}_n^-$$ C n - , but we assume that the family $${\mathcal {F}}$$ F is linearly orderable (for $$\textrm{LOC}_n^-$$ LOC n - ) or well-orderable (for $$\textrm{WOC}_n^-$$ WOC n - ). In the first part of this paper, for $$m,n\in \omega $$ m , n ∈ ω we will give a full characterization of when the implication $$\textrm{RC}_m\Rightarrow \textrm{WOC}_n^-$$ RC m ⇒ WOC n - holds in $${\textsf {ZF}}$$ 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 $$\textrm{RC}_5\Rightarrow \textrm{LOC}_5^-$$ RC 5 ⇒ LOC 5 - and that $$\textrm{RC}_6\Rightarrow \textrm{C}_3^-$$ RC 6 ⇒ C 3 - , answering two open questions from Halbeisen and Tachtsis (Arch Math Logik 59(5):583–606, 2020). Furthermore, we will show that $$\textrm{RC}_6\Rightarrow \textrm{C}_9^-$$ RC 6 ⇒ C 9 - and that $$\textrm{RC}_7\Rightarrow \textrm{LOC}_7^-$$ RC 7 ⇒ LOC 7 - .