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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":true},"canonical_record":{"source":{"id":"2604.14461","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.LO","submitted_at":"2026-04-15T22:39:21Z","cross_cats_sorted":[],"title_canon_sha256":"ba8b465fc4fc44e758c059c9ee56b5514e8428ae1ac841461b014e120bcc410f","abstract_canon_sha256":"f603a6c1546ac500bdc0509e852b8ae0ce2c9208c215e4d63b1fd14b100b6c9a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-04T01:08:50.138660Z","signature_b64":"prwrZKdN3Yk4TKimUJghbtvvlwnCuOjPLCLkDqocuhpMXcxKH6GdB1L07vqzkIit18euHDbQWgHoOf9QD07UDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dd09a5077af237a5a8f6e0883ae2d8b3d56d1b8cbd54080dae32b21735055825","last_reissued_at":"2026-06-04T01:08:50.138132Z","signature_status":"signed_v1","first_computed_at":"2026-06-04T01:08:50.138132Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $X$. 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