Cardinal characteristics and countable Borel equivalence relations

Mathematical Logic Quarterly 63 (3-4):211-227 (2017)
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Abstract

Boykin and Jackson recently introduced a property of countable Borel equivalence relations called Borel boundedness, which they showed is closely related to the union problem for hyperfinite equivalence relations. In this paper, we introduce a family of properties of countable Borel equivalence relations which correspond to combinatorial cardinal characteristics of the continuum in the same way that Borel boundedness corresponds to the bounding number. We analyze some of the basic behavior of these properties, showing, e.g., that the property corresponding to the splitting number coincides with smoothness. We then settle many of the implication relationships between the properties; these relationships turn out to be closely related to (but not the same as) the Borel Tukey ordering on cardinal characteristics.

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References found in this work

Countable borel equivalence relations.S. Jackson, A. S. Kechris & A. Louveau - 2002 - Journal of Mathematical Logic 2 (01):1-80.
Martin’s conjecture and strong ergodicity.Simon Thomas - 2009 - Archive for Mathematical Logic 48 (8):749-759.
No Borel Connections for the Unsplitting Relations.Heike Mildenberger - 2002 - Mathematical Logic Quarterly 48 (4):517-521.

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