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Logic"],"published-print":{"date-parts":[[2026,4]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We consider a large family of theories of equivalence relations, each with finitely many classes, and assuming the existence of an\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\omega $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>\u03c9<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    -Erd\u0151s cardinal, we determine which of these theories are Borel complete. We develop machinery, including\n                    <jats:italic>forbidding nested sequences<\/jats:italic>\n                    which implies a tight upper bound on Borel complexity, and\n                    <jats:italic>admitting cross-cutting absolutely indiscernible sets<\/jats:italic>\n                    which in our context implies Borel completeness.\n                  <\/jats:p>","DOI":"10.1007\/s00153-026-01007-5","type":"journal-article","created":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T18:35:00Z","timestamp":1772130900000},"page":"477-508","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Borel complexity of families of finite equivalence relations via large cardinals"],"prefix":"10.1007","volume":"65","author":[{"given":"Michael C.","family":"Laskowski","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Danielle S.","family":"Ulrich","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,2,26]]},"reference":[{"key":"1007_CR1","unstructured":"Allison, S.: Classification Strength of Polish Groups I: Involving $$S_\\infty $$, arXiv:2304.00139"},{"issue":"5","key":"1007_CR2","doi-asserted-by":"publisher","first-page":"3673","DOI":"10.1090\/tran\/6572","volume":"368","author":"JT Baldwin","year":"2016","unstructured":"Baldwin, J.T., Friedman, S.D., Koerwien, M., Laskowski, M.C.: Three red herrings around Vaught\u2019s conjecture. 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