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The saturation number $\\operatorname{sat}(\\mathcal{F},n)$ is the minimum number of edges in an $\\mathcal{F}$-saturated graph on $n$ vertices. We prove that there exists a finite family $\\mathcal{F}$ such that $\\operatorname{sat}(\\mathcal{F},n) / n^{r-1}$ does not tend to a limit. This settles a question of Pikhurko."},"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":false},"canonical_record":{"source":{"id":"1803.05799","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-03-15T15:18:22Z","cross_cats_sorted":[],"title_canon_sha256":"a26e98e86d13c0fbd9a171eacfd0b5b65fb31938c6bc433da7a42ca37f034fb9","abstract_canon_sha256":"692fc2394504c89a6810a93fc1f70ae51dce9d00d77a6db1f181273b81670446"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:30:47.899623Z","signature_b64":"PNglw0FwSu84/10JP0MDi9K/+sNjE+Uo1cNCbafbfhpoLawnralkq7wTkpelUcx/DGJQnCSRd8HAxJxQ+7SaDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cc1aaea0ccb4d602b13b012f1768d7bfdbf2fa0ee655df1a18f906ae711c2b1e","last_reissued_at":"2026-07-05T01:30:47.899131Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:30:47.899131Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hypergraph Saturation Irregularities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Natalie C. Behague","submitted_at":"2018-03-15T15:18:22Z","abstract_excerpt":"Let $\\mathcal{F}$ be a family of $r$-graphs. An $r$-graph $G$ is called $\\mathcal{F}$-saturated if it does not contain any members of $\\mathcal{F}$ but adding any edge creates a copy of some $r$-graph in $\\mathcal{F}$. The saturation number $\\operatorname{sat}(\\mathcal{F},n)$ is the minimum number of edges in an $\\mathcal{F}$-saturated graph on $n$ vertices. We prove that there exists a finite family $\\mathcal{F}$ such that $\\operatorname{sat}(\\mathcal{F},n) / n^{r-1}$ does not tend to a limit. 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