{"paper":{"title":"Small sets in dense pairs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Pantelis E. Eleftheriou","submitted_at":"2017-04-19T16:31:34Z","abstract_excerpt":"Let $\\widetilde{\\mathcal M}=\\langle \\mathcal M, P\\rangle$ be an expansion of an o-minimal structure $\\mathcal M$ by a dense set $P\\subseteq M$, such that three tameness conditions hold. We prove that the induced structure on $P$ by $\\mathcal M$ eliminates imaginaries. As a corollary, we obtain that every small set $X$ definable in $\\widetilde{\\mathcal M}$ can be definably embedded into some $P^l$, uniformly in parameters, settling a question from [10]. We verify the tameness conditions in three examples: dense pairs of real closed fields, expansions of $\\mathcal M$ by a dense independent set, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.05802","kind":"arxiv","version":4},"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"}