{"paper":{"title":"A universality theorem for allowable sequences with applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Keno Merckx, Udo Hoffmann","submitted_at":"2018-01-18T12:55:13Z","abstract_excerpt":"Order types are a well known abstraction of combinatorial properties of a point set. By Mn\\\"ev's universality theorem for each semi-algebraic set $V$ there is an order type with a realization space that is \\emph{stably equivalent} to $V$. We consider realization spaces of \\emph{allowable sequences}, a refinement of order types. We show that the realization spaces of allowable sequences are \\emph{universal} and consequently deciding the realizability is complete in the \\emph{existential theory of the reals} (\\ER). This result holds even if the realization space of the order type induced by the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.05992","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"}