{"paper":{"title":"Some applications of the real strict order property hierarchy","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Scott Mutchnik","submitted_at":"2026-06-27T05:31:35Z","abstract_excerpt":"We give applications of the properties $\\mathrm{NSOP}_{r}$ for non-integer values of $r$ to problems on the original hierarchy $\\mathrm{NSOP}_{n}$ for integer values of $n$.\n  We first show that the properties $\\mathrm{NSOP}_{r}$, previously defined for real values $r \\geq 3$, are even well-defined for real values $r \\geq 2$, showing that $\\mathrm{NSOP}_{2} \\subseteq \\mathrm{NSOP}_{r}$ for our original definition of $\\mathrm{NSOP}_{r}$ even when $2 < r < 3$. As a consequence, newness of all of the well-defined properties $\\mathrm{NSOP}_{r}$ for non-integer $r$ would negatively resolve the prob"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.28740","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.28740/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}