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We show that for the case $T=K_m$ and $\\chi(H)> m$ the behavior of $ex(G(n,p),K_m,H)$ depends strongly on the relation between $p$ and $m_2(H)=\\max_{H'\\subset H, |V(H')|'\\geq 3}\\left\\{ \\frac{e(H')-1}{v(H')-2} \\right\\}$.\n  When $m_2(H)> m_2(K_m)$ we prove that with high probability, depending on the value of $p$, either one can maintain almost all copies of $K_m$, or it is asymptotically best to take a $\\chi(H)-1$ partite"},"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":"1612.09143","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-29T13:49:27Z","cross_cats_sorted":[],"title_canon_sha256":"615ca875b92b94875d77e4ae8c37226dad854c0e2270e4fc33e887b6e5d6e761","abstract_canon_sha256":"bab34b71f432f9263e9a8acf395a8d07c491af49a8209d5355f2914d001d27c2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:30:17.283352Z","signature_b64":"UwnLjalGvvqHw3w2q3jGk8FOQ3mr7eenAd88M384YlReGqpgMVEL9UPrmDirBmtIhwHKGzC36DG2rlupdv/eDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"520198cfe3ccc26ebd3962233979c313b567a7909414c8fe4776cb4c7a2a157e","last_reissued_at":"2026-05-18T00:30:17.282500Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:30:17.282500Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Many cliques in $H$-free subgraphs of random graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexandr Kostochka, Clara Shikhelman, Noga Alon","submitted_at":"2016-12-29T13:49:27Z","abstract_excerpt":"For two fixed graphs $T$ and $H$ let $ex(G(n,p),T,H)$ be the random variable counting the maximum number of copies of $T$ in an $H$-free subgraph of the random graph $G(n,p)$. We show that for the case $T=K_m$ and $\\chi(H)> m$ the behavior of $ex(G(n,p),K_m,H)$ depends strongly on the relation between $p$ and $m_2(H)=\\max_{H'\\subset H, |V(H')|'\\geq 3}\\left\\{ \\frac{e(H')-1}{v(H')-2} \\right\\}$.\n  When $m_2(H)> m_2(K_m)$ we prove that with high probability, depending on the value of $p$, either one can maintain almost all copies of $K_m$, or it is asymptotically best to take a $\\chi(H)-1$ partite"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.09143","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":"1612.09143","created_at":"2026-05-18T00:30:17.282660+00:00"},{"alias_kind":"arxiv_version","alias_value":"1612.09143v2","created_at":"2026-05-18T00:30:17.282660+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.09143","created_at":"2026-05-18T00:30:17.282660+00:00"},{"alias_kind":"pith_short_12","alias_value":"KIAZRT7DZTBG","created_at":"2026-05-18T12:30:25.849896+00:00"},{"alias_kind":"pith_short_16","alias_value":"KIAZRT7DZTBG5PJZ","created_at":"2026-05-18T12:30:25.849896+00:00"},{"alias_kind":"pith_short_8","alias_value":"KIAZRT7D","created_at":"2026-05-18T12:30:25.849896+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/KIAZRT7DZTBG5PJZMIRTS6ODCO","json":"https://pith.science/pith/KIAZRT7DZTBG5PJZMIRTS6ODCO.json","graph_json":"https://pith.science/api/pith-number/KIAZRT7DZTBG5PJZMIRTS6ODCO/graph.json","events_json":"https://pith.science/api/pith-number/KIAZRT7DZTBG5PJZMIRTS6ODCO/events.json","paper":"https://pith.science/paper/KIAZRT7D"},"agent_actions":{"view_html":"https://pith.science/pith/KIAZRT7DZTBG5PJZMIRTS6ODCO","download_json":"https://pith.science/pith/KIAZRT7DZTBG5PJZMIRTS6ODCO.json","view_paper":"https://pith.science/paper/KIAZRT7D","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1612.09143&json=true","fetch_graph":"https://pith.science/api/pith-number/KIAZRT7DZTBG5PJZMIRTS6ODCO/graph.json","fetch_events":"https://pith.science/api/pith-number/KIAZRT7DZTBG5PJZMIRTS6ODCO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KIAZRT7DZTBG5PJZMIRTS6ODCO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KIAZRT7DZTBG5PJZMIRTS6ODCO/action/storage_attestation","attest_author":"https://pith.science/pith/KIAZRT7DZTBG5PJZMIRTS6ODCO/action/author_attestation","sign_citation":"https://pith.science/pith/KIAZRT7DZTBG5PJZMIRTS6ODCO/action/citation_signature","submit_replication":"https://pith.science/pith/KIAZRT7DZTBG5PJZMIRTS6ODCO/action/replication_record"}},"created_at":"2026-05-18T00:30:17.282660+00:00","updated_at":"2026-05-18T00:30:17.282660+00:00"}