Cardinal characteristics and countable Borel equivalence relations

TL;DR

This paper explores new properties of countable Borel equivalence relations linked to continuum cardinal characteristics, analyzing their behavior and relationships, and relating them to Borel Tukey orderings.

Contribution

It introduces a family of properties of countable Borel equivalence relations corresponding to continuum cardinal characteristics, expanding understanding of their structure and interrelations.

Findings

The property for the splitting number $rak s$ coincides with smoothness.

Many implication relationships between properties are established.

Relationships are connected to Borel Tukey ordering on cardinal characteristics.

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 $\mathfrak b$. We analyze some of the basic behavior of these properties, showing for instance that the property corresponding to the splitting number $\mathfrak s$ 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.