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We prove that if $p n^{k-1}/\\log n$ tends to infinity with $n$ then $$\\lim_{\\substack{n\\to \\infty 2(k-1) |n}}\\Pr(H_{n,p;k}\\ contains\\ a\\ loose\\ Hamilton\\ cycle)=1.$$ This is asymptotically best possible."},"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":"1006.1909","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-06-09T21:07:47Z","cross_cats_sorted":[],"title_canon_sha256":"2ec66b907b612d9c6466cd37a199060660ac54b64d4863a2b3d8771f09c67ca3","abstract_canon_sha256":"eaf582743c6bc69b43d0fc02cabdc28e4037700a6d76fdd4e7bf1ab9adaed019"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:28:08.207860Z","signature_b64":"o30+4IhJ71xacJJdMLOMOvf1oSxXO3YXJ8vk4peTxD/wVzHfrMq2kwUW/yZIkOyoPoNj3ukuj9hYtzciiTV3BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2f73b47f1451c95b0149070977e003b479bc160ee001bccca4e01cda377cdc2f","last_reissued_at":"2026-05-18T04:28:08.207179Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:28:08.207179Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Loose Hamilton Cycles in Random Uniform Hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alan Frieze, Andrzej Dudek","submitted_at":"2010-06-09T21:07:47Z","abstract_excerpt":"In the random hypergraph $H_{n,p;k}$ each possible $k$-tuple appears independently with probability $p$. 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