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It is a much studied and surprisingly difficult problem to understand the upper tail of the distribution of $\\xi_H$, for example, to estimate \\begin{equation*}\n  \\mathbb{P}(\\xi_H > 2 \\mathbb{E}\\xi_H). \\end{equation*} The best known result for general $H$ and $p$ is due to Janson, Oleszkiewicz, and Ruci\\'nski, who, in 2004, proved \\begin{align}\\label{a:JOR} \\exp[-O_{H, \\eta}(M_H(n,p) \\ln(1/p))]&<\\mathbb{P}(\\xi_H > (1+\\eta)\\mat"},"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":"1903.07488","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-03-18T14:49:22Z","cross_cats_sorted":[],"title_canon_sha256":"ca2214986e545c3f2fc14cd7e3ecb3211a6abc642ff564ffe2b8f36e3ceb4e72","abstract_canon_sha256":"ebcd4257d11f27a0fcbfc7a752d347488e9b651c476bea8eb0ed69533940086b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:49:38.495691Z","signature_b64":"CqB6PfRvS8NaMZWgni68oYE4q00E6BJ5N/3Hp6ohdcukrR6MC95CZD9ZJWgL39GsYa+o5M4jGAny6NX37xX4Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f8dfe30434a92d2f8d1a92c3048636b310d1f01429d680b292473d019e3e841b","last_reissued_at":"2026-05-17T23:49:38.494923Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:49:38.494923Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Upper tail bounds for cycles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Abigail Raz","submitted_at":"2019-03-18T14:49:22Z","abstract_excerpt":"This paper examines bounds on upper tails for cycle counts in $G_{n,p}$. For a fixed graph $H$ define $\\xi_H= \\xi_H^{n,p}$ to be the number of copies of $H$ in $G_{n,p}$. 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