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<p>We consider a large family of theories of equivalence relations, each with finitely many classes, and assuming the existence of an <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\omega \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation>-Erdős cardinal, we determine which of these theories are Borel complete. We develop machinery, including <i>forbidding nested sequences</i> which implies a tight upper bound on Borel complexity, and <i>admitting cross-cutting absolutely indiscernible sets</i> which in our context implies Borel completeness.</p>

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Borel complexity of families of finite equivalence relations via large cardinals

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月26日

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  • Michael C. Laskowski,
  • Danielle S. Ulrich

摘要

We consider a large family of theories of equivalence relations, each with finitely many classes, and assuming the existence of an \(\omega \) 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.