{"paper":{"title":"A Dividing Line Within Simple Unstable Theories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"M. Malliaris, S. Shelah","submitted_at":"2012-08-10T10:55:19Z","abstract_excerpt":"We give the first (ZFC) dividing line in Keisler's order among the unstable theories, specifically among the simple unstable theories. That is, for any infinite cardinal $\\lambda$ for which there is $\\mu < \\lambda \\leq 2^\\mu$, we construct a regular ultrafilter D on $\\lambda$ such that (i) for any model $M$ of a stable theory or of the random graph, $M^\\lambda/D$ is $\\lambda^+$-saturated but (ii) if $Th(N)$ is not simple or not low then $N^\\lambda/D$ is not $\\lambda^+$-saturated. The non-saturation result relies on the notion of flexible ultrafilters. To prove the saturation result we develop "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.2140","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"}