  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.
