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The $k$-uniform saturation number of Berge-$F$, $\\mathrm{sat}_k(n,\\text{Berge-}F)$ is the fewest number of edges in a Berge-$F$-saturated $k$-uniform hypergraph on $n$ vertices. We show that $\\mathrm{sat}_k(n,\\text{Berge-}F) = O(n)$ for all graphs $F$ and uniformities $3\\leq k\\leq 5$, partially answering a conjecture o"},"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":"1807.06947","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-18T14:01:55Z","cross_cats_sorted":[],"title_canon_sha256":"9f727aae2279290264c2799c614dae4ded4ce76e2603fed721e1c287c64261e0","abstract_canon_sha256":"2a8a64fedf8eabeff29306b1078ea3e98788597209a1b0eed1f1776a156c06ca"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:10:25.254467Z","signature_b64":"NTQI2cImdreOomvVof2f6Y0FBbrc4+XLUnfZNeKqx5dv7IXQeMXl3U9t9Vsb+xtRuNuLBjqqCOxT9tO1LnwLDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fdb7a6cb3b4f774a2e50da6839f03d55d0b1a3be7926abacfe03d111733d09ed","last_reissued_at":"2026-05-18T00:10:25.253902Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:10:25.253902Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Linearity of Saturation for Berge Hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Abhishek Methuku, D\\'aniel Gerbner, Michael Tait, Sean English","submitted_at":"2018-07-18T14:01:55Z","abstract_excerpt":"For a graph $F$, we say a hypergraph $H$ is Berge-$F$ if it can be obtained from $F$ be replacing each edge of $F$ with a hyperedge containing it. 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