{"paper":{"title":"A rank function for Fra\\\"{\\i}ss\\'{e} classes and the rank property","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"Certain Fraïssé classes realize every countable ordinal as a value of the rank function measuring distance from universality.","cross_cats":[],"primary_cat":"math.LO","authors_text":"Carlos L\\'opez-Callejas, Jareb Navarro-Castillo","submitted_at":"2026-04-15T22:39:21Z","abstract_excerpt":"Given a hereditary class $\\mathcal{F}$ of finite relational structures, the rank function $\\mathsf{rk}:\\sigma\\mathcal{F}\\to\\omega_1\\cup\\{\\infty\\}$, introduced by Kubi\\'{s} and Shelah, measures how far a countable structure is from being universal within its class: $\\mathsf{rk}(X)=\\infty$ if and only if the Fra\\\"{\\i}ss\\'{e} limit embeds into $X$. We say that $\\mathcal{F}$ has the Rank Property (RP) if every countable ordinal is realized as the rank of some $X\\in\\sigma\\mathcal{F}$.\n  We develop the basic theory of the rank function and establish RP for three families of classes: those satisfying"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We establish RP for three families of classes: those satisfying the free amalgamation property and the full extension property (covering graphs, hypergraphs, and many others); finite tournaments; and finite linear orders. For the latter, we compute the rank of every countable ordinal: if ω^{β₁}·c₁ is the leading Cantor normal form term of α≥ω, then rk(α)=ω·β₁ + ⌊log₂ c₁⌋.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The hereditary classes under consideration satisfy the free amalgamation property together with the full extension property, or consist exactly of finite tournaments or finite linear orders; the rank function is assumed to be well-defined on σF as introduced by Kubiš and Shelah.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Introduces theory of a rank function measuring distance from Fraïssé universality and proves it realizes every countable ordinal for free-amalgamation classes, tournaments, and linear orders with explicit computation.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Certain Fraïssé classes realize every countable ordinal as a value of the rank function measuring distance from universality.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"4148b18f2071f610be0cb0336d24ffd8351fe63a52aa61023d31c8b0e40527da"},"source":{"id":"2604.14461","kind":"arxiv","version":3},"verdict":{"id":"d37dc343-3b25-4536-ac5a-f020bd40c41f","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-13T07:56:06.368970Z","strongest_claim":"We establish RP for three families of classes: those satisfying the free amalgamation property and the full extension property (covering graphs, hypergraphs, and many others); finite tournaments; and finite linear orders. For the latter, we compute the rank of every countable ordinal: if ω^{β₁}·c₁ is the leading Cantor normal form term of α≥ω, then rk(α)=ω·β₁ + ⌊log₂ c₁⌋.","one_line_summary":"Introduces theory of a rank function measuring distance from Fraïssé universality and proves it realizes every countable ordinal for free-amalgamation classes, tournaments, and linear orders with explicit computation.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The hereditary classes under consideration satisfy the free amalgamation property together with the full extension property, or consist exactly of finite tournaments or finite linear orders; the rank function is assumed to be well-defined on σF as introduced by Kubiš and Shelah.","pith_extraction_headline":"Certain Fraïssé classes realize every countable ordinal as a value of the rank function measuring distance from universality."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.14461/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":1,"snapshot_sha256":"379cc12930cd926264a0ddd1129e8081ef3e31c714844edee1db075843224b66"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}