Borel complexity of families of finite equivalence relations via large cardinals

Abstract We consider a large family of theories of equivalence relations, each with finitely many classes, and assuming the existence of an $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> -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.