  We consider a large family of theories of equivalence relations, each with
finitely many classes, and assuming the existence of an $\omega$-Erdos
cardinal, we determine which of these theories are Borel complete. We develop
machinery, including {\em forbidding nested sequences} which implies a tight
upper bound on Borel complexity, and {\em admitting cross-cutting absolutely
indiscernible sets} which in our context implies Borel completeness. In the
Appendix we classify the reducts of theories of refining equivalence relations,
possibly with infinite splitting.
