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We show that if $0<p<\\gamma<1$ then $\\omega_{\\gamma}(G_{n,p})$ is concentrated on a set of two integers. More precisely, with $\\alpha(\\gamma,p)=\\gamma\\log\\frac{\\gamma}{p}+(1-\\gamma)\\log\\frac{1-\\gamma}{1-p}$, we show that $\\omega_{\\gamma}(G_{n,p})$ is one of the two integers closest to $\\frac{2}{\\alpha(\\gamma,p)}\\big(\\log n-\\log\\log n+\\log\\frac{e\\alpha(\\gamma,p)}{2}\\big)+\\frac{1}{2}$, with high probabi"},"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":"1803.10349","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-03-27T22:39:17Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"6db84ed6a2ef7b24bed2b97a9ff662f1b2825451a9dd0498df6fc19e5c6716ad","abstract_canon_sha256":"cb4ca4ea7059f0883b94982647dc6cf8bb2c9b8d95db601a64a62249d66ce0e2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:19:55.704442Z","signature_b64":"ol+jgrVVE3pfIyJ4Xfb4llKHDl3nsHh6DRjHy6zFqU+D7yZuTsp3/llvgTbWNGE9xC9xz4i0SOC6YlFDzHvdCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b8a3a00a34bedaf883059938721f352101259e84c55d28a45acb87f920435efe","last_reissued_at":"2026-05-18T00:19:55.703839Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:19:55.703839Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Dense Subgraphs in Random Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Alexander Veremyev, B\\'ela Bollob\\'as, Julian Sahasrabudhe, Paul Balister","submitted_at":"2018-03-27T22:39:17Z","abstract_excerpt":"For a constant $\\gamma \\in[0,1]$ and a graph $G$, let $\\omega_{\\gamma}(G)$ be the largest integer $k$ for which there exists a $k$-vertex subgraph of $G$ with at least $\\gamma\\binom{k}{2}$ edges. 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