Abstract
For an integer \(n\ge 2\), Ramsey Choice\(\mathsf {RC}_{n}\) is the weak choice principle “every infinite setxhas an infinite subset y such that\([y]^{n}\) (the set of alln-element subsets of y) has a choice function”, and \(\mathsf {C}_{n}^{-}\) is the weak choice principle “every infinite family of n-element sets has an infinite subfamily with a choice function”. In 1995, Montenegro showed that for \(n=2,3,4\), \(\mathsf {RC}_{n}\rightarrow \mathsf {C}_{n}^{-}\). However, the question of whether or not \(\mathsf {RC}_{n}\rightarrow \mathsf {C}_{n}^{-}\) for \(n\ge 5\) is still open. In general, for distinct \(m,n\ge 2\), not even the status of “\(\mathsf {RC}_{n}\rightarrow \mathsf {C}_{m}^{-}\)” or “\(\mathsf {RC}_{n}\rightarrow \mathsf {RC}_{m}\)” is known. In this paper, we provide partial answers to the above open problems and among other results, we establish the following:
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1.
For every integer \(n\ge 2\), if \(\mathsf {RC}_{i}\) is true for all integers i with \(2\le i\le n\), then \(\mathsf {C}_{i}^{-}\) is true for all integers i with \(2\le i\le n\).
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2.
If \(m,n\ge 2\) are any integers such that for some prime p we have \(p\not \mid m\) and \(p\mid n\), then in \(\mathsf {ZF}\): \(\mathsf {RC}_{m}\nrightarrow \mathsf {RC}_{n}\) and \(\mathsf {RC}_{m}\nrightarrow \mathsf {C}_{n}^{-}\).
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3.
For \(n=2,3\), \(\mathsf {RC}_{5}\)\(+\)\(\mathsf {C}_{n}^{-}\) implies \(\mathsf {C}_{5}^{-}\), and \(\mathsf {RC}_{5}\) implies neither \(\mathsf {C}_{2}^{-}\) nor \(\mathsf {C}_{3}^{-}\) in \(\mathsf {ZF}\).
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4.
For every integer \(k\ge 2\), \(\mathsf {RC}_{2k}\) implies “every infinite linearly orderable family of k-element sets has a partial Kinna–Wagner selection function” and the latter implication is not reversible in \(\mathsf {ZF}\) (for any \(k\in \omega \backslash \{0,1\}\)). In particular, \(\mathsf {RC}_{6}\) strictly implies “every infinite linearly orderable family of 3-element sets has a partial choice function”.
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5.
The Chain-AntiChain Principle (“every infinite partially ordered set has either an infinite chain or an infinite anti-chain”) implies neither \(\mathsf {RC}_{n}\) nor \(\mathsf {C}_{n}^{-}\) in \(\mathsf {ZF}\), for every integer \(n\ge 2\).
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Acknowledgements
We would like to thank the anonymous referee for various comments and suggestions which substantially improved both the mathematical quality and the exposition of this paper.
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Halbeisen, L., Tachtsis, E. On Ramsey choice and partial choice for infinite families of n-element sets. Arch. Math. Logic 59, 583–606 (2020). https://doi.org/10.1007/s00153-019-00705-7
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DOI: https://doi.org/10.1007/s00153-019-00705-7
Keywords
- Axiom of Choice
- Weak forms of the Axiom of Choice
- Ramsey Choice
- Partial Choice for infinite families of n-element sets
- Ramsey’s Theorem
- Chain-AntiChain Principle
- Fraenkel–Mostowski permutation models of \(\mathsf {ZFA}+\lnot \mathsf {AC}\)
- Pincus’ Transfer Theorems