Borel complexity of families of finite equivalence relations via large cardinals

TL;DR

This paper investigates the Borel complexity of families of finite equivalence relation theories under large cardinal assumptions, introducing new methods to classify their complexity and establishing criteria for Borel completeness.

Contribution

It develops novel techniques like forbidding nested sequences and cross-cutting indiscernible sets to determine Borel completeness of equivalence relation theories.

Findings

Identifies conditions for Borel completeness of finite equivalence relation theories.

Introduces machinery to bound Borel complexity using large cardinal assumptions.

Classifies reducts of theories of refining equivalence relations.

Abstract

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.