On Ramsey choice and partial choice for infinite families of n-element sets
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2020-08
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Journal Article
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Abstract
For an integer n≥ 2 , Ramsey ChoiceRCn 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 Cn- 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 , RCn→Cn-. However, the question of whether or not RCn→Cn- for n≥ 5 is still open. In general, for distinct m, n≥ 2 , not even the status of “RCn→Cm-” or “RCn→ RCm” is known. In this paper, we provide partial answers to the above open problems and among other results, we establish the following:
1.For every integer n≥ 2 , if RCi is true for all integers i with 2 ≤ i≤ n, then Ci- is true for all integers i with 2 ≤ i≤ n.
2.If m, n≥ 2 are any integers such that for some prime p we have p∤ m and p∣ n, then in ZF: RCm↛ RCn and RCm↛Cn-.
3.For n= 2 , 3 , RC5+ Cn- implies C5-, and RC5 implies neither C2- nor C3- in ZF.
4.For every integer k≥ 2 , RC2 k 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 ZF (for any k∈ ω\ { 0 , 1 }). In particular, RC6 strictly implies “every infinite linearly orderable family of 3-element sets has a partial choice function”.
5.The Chain-AntiChain Principle (“every infinite partially ordered set has either an infinite chain or an infinite anti-chain”) implies neither RCn nor Cn- in ZF, for every integer n≥ 2.
© 2019 Springer.
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59 (5)
Pages / Article No.
583 - 606
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Springer
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Subject
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 ZFA AC; Pincus' Transfer Theorems
Organisational unit
03874 - Hungerbühler, Norbert / Hungerbühler, Norbert