{"paper":{"title":"Gluing Posets and the Dichotomy of Poset Saturation Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Maria-Romina Ivan, Sean Jaffe","submitted_at":"2025-03-15T18:17:15Z","abstract_excerpt":"Given a finite poset $\\mathcal P$, we say that a family $\\mathcal F$ of subsets of $[n]$ is $\\mathcal P$-saturated if $\\mathcal F$ does not contain an induced copy of $\\mathcal P$, but adding any other set to $\\mathcal F$ creates an induced copy of $\\mathcal P$. The saturation number of $\\mathcal P$ is the size of the smallest $\\mathcal P$-saturated family with ground set $[n]$. The saturation number for posets is known to exhibit a dichotomy: it is either bounded or it has at least $\\sqrt n$ rate of growth. Determining which posets have bounded saturation number is a major open problem.\n  In "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2503.12223","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2503.12223/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"}