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We give an asymptotic formula for the number $C_r(n,m)$ of connected $r$-uniform hypergraphs on $[n]$ with $m$ edges, whenever $r\\ge 3$ is fixed and $m=m(n)$ with $m/n\\to\\infty$, i.e., the average degree tends to infinity. This complements recent results of Behrisch, Coja-Oghlan and Kang (the case $m=n/(r-1)+\\Theta(n)$) and the present authors (the case $m=n/(r-1)+o(n)$, i.e., `nullity' or `excess' "},"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":"1511.04739","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-11-15T17:56:47Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"92145809d35542f61c979e36521f242704580f898e976e01c8c080083a737465","abstract_canon_sha256":"37599cad7735518905c4400e7fe099d44f25cc3fbf3569619af83a7bc2fa4178"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:01:44.095551Z","signature_b64":"plvTe+Z21JtumAki3nsdMSL7BZKXNJYdRIMULdomnu1XyVuBBiaSontxo/8QxqvUjFebDWimSgt1tedboqBNAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6f591a5e042321b5c28d2390a8de7a3be18fd8c5445ecaa33e8b9e90dda7d1e8","last_reissued_at":"2026-05-18T00:01:44.095036Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:01:44.095036Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Counting dense connected hypergraphs via the probabilistic method","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"B\\'ela Bollob\\'as, Oliver Riordan","submitted_at":"2015-11-15T17:56:47Z","abstract_excerpt":"In 1990 Bender, Canfield and McKay gave an asymptotic formula for the number of connected graphs on $[n]=\\{1,2,\\ldots,n\\}$ with $m$ edges, whenever $n\\to\\infty$ and $n-1\\le m=m(n)\\le \\binom{n}{2}$. We give an asymptotic formula for the number $C_r(n,m)$ of connected $r$-uniform hypergraphs on $[n]$ with $m$ edges, whenever $r\\ge 3$ is fixed and $m=m(n)$ with $m/n\\to\\infty$, i.e., the average degree tends to infinity. This complements recent results of Behrisch, Coja-Oghlan and Kang (the case $m=n/(r-1)+\\Theta(n)$) and the present authors (the case $m=n/(r-1)+o(n)$, i.e., `nullity' or `excess' "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.04739","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1511.04739","created_at":"2026-05-18T00:01:44.095114+00:00"},{"alias_kind":"arxiv_version","alias_value":"1511.04739v2","created_at":"2026-05-18T00:01:44.095114+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.04739","created_at":"2026-05-18T00:01:44.095114+00:00"},{"alias_kind":"pith_short_12","alias_value":"N5MRUXQEEMQ3","created_at":"2026-05-18T12:29:32.376354+00:00"},{"alias_kind":"pith_short_16","alias_value":"N5MRUXQEEMQ3LQUN","created_at":"2026-05-18T12:29:32.376354+00:00"},{"alias_kind":"pith_short_8","alias_value":"N5MRUXQE","created_at":"2026-05-18T12:29:32.376354+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/N5MRUXQEEMQ3LQUNEOIKRXT2HP","json":"https://pith.science/pith/N5MRUXQEEMQ3LQUNEOIKRXT2HP.json","graph_json":"https://pith.science/api/pith-number/N5MRUXQEEMQ3LQUNEOIKRXT2HP/graph.json","events_json":"https://pith.science/api/pith-number/N5MRUXQEEMQ3LQUNEOIKRXT2HP/events.json","paper":"https://pith.science/paper/N5MRUXQE"},"agent_actions":{"view_html":"https://pith.science/pith/N5MRUXQEEMQ3LQUNEOIKRXT2HP","download_json":"https://pith.science/pith/N5MRUXQEEMQ3LQUNEOIKRXT2HP.json","view_paper":"https://pith.science/paper/N5MRUXQE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1511.04739&json=true","fetch_graph":"https://pith.science/api/pith-number/N5MRUXQEEMQ3LQUNEOIKRXT2HP/graph.json","fetch_events":"https://pith.science/api/pith-number/N5MRUXQEEMQ3LQUNEOIKRXT2HP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/N5MRUXQEEMQ3LQUNEOIKRXT2HP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/N5MRUXQEEMQ3LQUNEOIKRXT2HP/action/storage_attestation","attest_author":"https://pith.science/pith/N5MRUXQEEMQ3LQUNEOIKRXT2HP/action/author_attestation","sign_citation":"https://pith.science/pith/N5MRUXQEEMQ3LQUNEOIKRXT2HP/action/citation_signature","submit_replication":"https://pith.science/pith/N5MRUXQEEMQ3LQUNEOIKRXT2HP/action/replication_record"}},"created_at":"2026-05-18T00:01:44.095114+00:00","updated_at":"2026-05-18T00:01:44.095114+00:00"}