{"paper":{"title":"Dichotomy Theorems for Families of Non-Cofinal Essential Complexity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GN"],"primary_cat":"math.LO","authors_text":"Benjamin D. Miller, Dominique Lecomte (IMJ), John D. Clemens","submitted_at":"2014-12-30T17:00:21Z","abstract_excerpt":"We prove that for every Borel equivalence relation $E$, either $E$ is Borel reducible to $\\mathbb{E}\\_0$, or the family of Borel equivalence relations incompatible with $E$ has cofinal essential complexity. It follows that if $F$ is a Borel equivalence relation and $\\cal F$ is a family of Borel equivalence relations of non-cofinal essential complexity which together satisfy the dichotomy that for every Borel equivalence relation $E$, either $E\\in {\\cal F}$ or $F$ is Borel reducible to $E$, then $\\cal F$ consists solely of smooth equivalence relations, thus the dichotomy is equivalent to a know"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.8684","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}