On Ramsey choice and partial choice for infinite families of n -element sets

Archive for Mathematical Logic 59 (5-6):583-606 (2020)
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Abstract

For an integer \, Ramsey Choice\ is the weak choice principle “every infinite setxhas an infinite subset y such that\ has a choice function”, and \ 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 \, \. However, the question of whether or not \ for \ is still open. In general, for distinct \, not even the status of “\” or “\” 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 \, if \ is true for all integers i with \, then \ is true for all integers i with \.2.If \ are any integers such that for some prime p we have \ and \, then in \: \ and \.3.For \, \\\ implies \, and \ implies neither \ nor \ in \.4.For every integer \, \ 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 \ ). In particular, \ strictly implies “every infinite linearly orderable family of 3-element sets has a partial choice function”.5.The Chain-AntiChain Principle implies neither \ nor \ in \, for every integer \.

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The Relation Between Two Diminished Choice Principles.Salome Schumacher - 2021 - Journal of Symbolic Logic 86 (1):415-432.

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References found in this work

Ramsey's theorem in the hierarchy of choice principles.Andreas Blass - 1977 - Journal of Symbolic Logic 42 (3):387-390.
The independence of Ramsey's theorem.E. M. Kleinberg - 1969 - Journal of Symbolic Logic 34 (2):205-206.
Finite axioms of choice.John Truss - 1973 - Annals of Mathematical Logic 6 (2):147.

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