Borel complexity of families of finite equivalence relations via large cardinals

We consider a large family of theories of equivalence relations, each with finitely many classes, and assuming the existence of an ω ω -Erdős cardinal, we determine which of these theories are Borel complete. We develop machinery, including forbidding nested sequences which implies a tight upper bound on Borel complexity, and admitting cross-cutting absolutely indiscernible sets which in our context implies Borel completeness.