Some implications of Ramsey choice for families of \(n\)-element sets (Q6103508)

In the paper under review, the authors substantially extend the research initiated by \textit{C. H. Montenegro} [Lect. Notes Pure Appl. Math. 203, 57--60 (1999; Zbl 0933.03064)] and continued by \textit{L. Halbeisen} and the reviewer [Arch. Math. Logic 59, No. 5--6, 583--606 (2020; Zbl 1472.03056)] on the deductive strength of the weak choice principles \(\mathrm{RC}_{n}\) (where \(n\in\omega\setminus\{0,1\}\)), which is ``For every infinite set \(X\), there is an infinite set \(Y\subseteq X\) with a choice function on \([Y]^{n}=\{z:z\subseteq Y\wedge |z|=n\}\)''; \(\mathrm{C}_{n}^{-}\), which is ``For every infinite family \(\mathcal{F}\) of \(n\)-element sets, there is an infinite family \(\mathcal{G}\subseteq\mathcal{F}\) with a choice function''; \(\mathrm{LOC}_{n}^{-}\) and \(\mathrm{WOC}_{n}^{-}\), which are the same as \(\mathrm{C}_{n}^{-}\) with the exception of \(\mathcal{F}\) being linearly orderable for \(\mathrm{LOC}_{n}^{-}\) and well orderable for \(\mathrm{WOC}_{n}^{-}\) (also see [\textit{S. Schumacher}, J. Symb. Log. 86, No. 1, 415--432 (2021; Zbl 1487.03059); \textit{L. Halbeisen} et al., ``A new weak choice principle'', Preprint, \url{arXiv:2101.07840}] for some closely related results). In the first part of the paper, the authors determine the values \(m,n\in\omega\setminus\{0,1\}\) for which the implication \(\mathrm{RC}_{m}\Rightarrow\mathrm{WOC}_{n}^{-}\) is provable in \(\mathrm{ZF}\) (i.e., provable without invoking any form of choice). In particular, they establish that, for every \(m,n\in\omega\setminus\{0,1\}\), ``\(\mathrm{RC}_{m}\Rightarrow\mathrm{WOC}_{n}^{-}\)'' is provable in \(\mathrm{ZF}\) if and only if the following condition holds: Whenever \(n\) can be written as \(n=\sum_{i<k}a_{i}p_{i}\), where, for every \(i<k\), \(p_{i}\) is a prime number and \(a_{i}\in\omega\setminus\{0\}\), then there exist \(b_{i}\in\omega\), \(i<k\), with \(m=\sum_{i<k}b_{i}p_{i}\). For the proof of ``\(\Rightarrow\)'' of the latter equivalence, it is shown that for all \(m,n\in\omega\setminus\{0,1\}\) which do not satisfy the previous condition, ``\(\mathrm{RC}_{m}\wedge\neg\mathrm{WOC}_{n}^{-}\)'' is relatively consistent with \(\mathrm{ZF}\). This is accomplished via the construction of suitable permutation models of \(\mathrm{ZFA}\) (i.e., Zermelo-Fraenkel set theory with atoms) which satisfy \(\mathrm{RC}_{m}\wedge\neg\mathrm{WOC}_{n}^{-}\) and then using a transfer theorem of Pincus for the transfer to \(\mathrm{ZF}\). In the second part of the paper, the authors establish the following in \(\mathrm{ZF}\): \begin{itemize} \item[(1)] \(\mathrm{RC}_{6}\Rightarrow\mathrm{C}_{3}^{-}\); \item[(2)] \(\mathrm{RC}_{6}\Rightarrow\mathrm{C}_{9}^{-}\); \item[(3)] \(\mathrm{RC}_{5}\Rightarrow\mathrm{LOC}_{5}^{-}\); \item[(4)] \(\mathrm{RC}_{7}\Rightarrow\mathrm{LOC}_{7}^{-}\). \end{itemize} (1) and (3) answer corresponding open questions from the paper by Halbeisen and the reviewer [loc. cit]. The paper concludes with a list of open questions. For example, Montenegro had shown that in \(\mathrm{ZF}\), \(\mathrm{RC}_{n}\Rightarrow\mathrm{C}_{n}^{-}\) for every \(n\in\{2,3,4\}\). However, whether or not this implication holds for any other \(n\in\omega\setminus\{0,1\}\) is still a challenging open problem (Question 1 of Section 4 of this paper).