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- W3125334583 abstract "Let $ninomega$. The weak choice principle $operatorname{RC}_n$ states that for every infinite set $x$ there is an infinite subset $ysubseteq x$ with a choice function on $[y]^n:={zsubseteq ymid lvert zrvert =n}$. $operatorname{C}_n^-$ states that for every infinite family of $n$-element sets, there is an infinite subfamily $mathcal{G}subseteqmathcal{F}$ 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 $operatorname{RC}_mRightarrow operatorname{WOC}_n^-$ with $m,ninomega$ 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 $operatorname{RC}_5Rightarrow operatorname{LOC}_5^-$ and $operatorname{RC}_6Rightarrow operatorname{C}_3^-$, answering two open questions from Halbeisen and Tachtsis. Furthermore we will show that $operatorname{RC}_6Rightarrow operatorname{C}_9^-$ and $operatorname{RC}_7Rightarrow operatorname{LOC}_7^-$." @default.
- W3125334583 created "2021-02-01" @default.
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- W3125334583 date "2021-01-18" @default.
- W3125334583 modified "2023-09-27" @default.
- W3125334583 title "Some implications of Ramsey Choice for n-element sets" @default.
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