Families of Borel equivalence relations and quasiorders that are cofinal with respect to the Borel reducibility ordering, ≤B, are constructed. There is an analytic ideal on ω generating a complete analytic equivalence relation and any Borel equivalence relation reduces to one generated by a Borel ideal. Several Borel equivalence relations, among them Lipschitz isomorphism of compact metric spaces, are shown to be Kσ complete.
"Cofinal families of Borel equivalence relations and quasiorders." J. Symbolic Logic 70 (4) 1325 - 1340, December 2005. https://doi.org/10.2178/jsl/1129642127