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Let $g(n,h) = f(n,h) + \\sum_{i=1}^h g(a_i,h)$. It is proved that for almost all graphs $H$ on $h$ vertices it holds that $i_H(n)=g(n,h)$ for all $n \\le 2^{\\sqrt{h}}$. More precisely, we define an explicit graph property ${\\cal P}_h$ which, when satisfied by $H$, guarantees that $i_H(n)=g(n,h)$ for a"},"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":"1801.01047","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-01-03T15:22:10Z","cross_cats_sorted":[],"title_canon_sha256":"3247960c7c0944fc694f59e40e60afa014992633af46d1193da04473cc6ae7e4","abstract_canon_sha256":"d8a6722542488de3c7f37b2fae2f393762d3b800b5c225221e1f1ad991162150"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:26:07.240878Z","signature_b64":"C5NDudfsqPfrz9sZrFjUJ/5yO2j46KYwT59mlTuit1JnwAfk57rTktYzB41heLD9sNWcbpj0EDPk/OMPaZPVDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7e6ea05f3de5287a7f52dd26d17ad0a36d431c237e40da0a2ca62d1b631f3bf6","last_reissued_at":"2026-05-18T00:26:07.240272Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:26:07.240272Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the exact maximum induced density of almost all graphs and their inducibility","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Raphael Yuster","submitted_at":"2018-01-03T15:22:10Z","abstract_excerpt":"Let $H$ be a graph on $h$ vertices. The number of induced copies of $H$ in a graph $G$ is denoted by $i_H(G)$. Let $i_H(n)$ denote the maximum of $i_H(G)$ taken over all graphs $G$ with $n$ vertices.\n  Let $f(n,h) = \\Pi_{i}^h a_i$ where $\\sum_{i=1}^h a_i = n$ and the $a_i$ are as equal as possible. Let $g(n,h) = f(n,h) + \\sum_{i=1}^h g(a_i,h)$. It is proved that for almost all graphs $H$ on $h$ vertices it holds that $i_H(n)=g(n,h)$ for all $n \\le 2^{\\sqrt{h}}$. More precisely, we define an explicit graph property ${\\cal P}_h$ which, when satisfied by $H$, guarantees that $i_H(n)=g(n,h)$ for a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.01047","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":"1801.01047","created_at":"2026-05-18T00:26:07.240366+00:00"},{"alias_kind":"arxiv_version","alias_value":"1801.01047v2","created_at":"2026-05-18T00:26:07.240366+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.01047","created_at":"2026-05-18T00:26:07.240366+00:00"},{"alias_kind":"pith_short_12","alias_value":"PZXKAXZ54UUH","created_at":"2026-05-18T12:32:46.962924+00:00"},{"alias_kind":"pith_short_16","alias_value":"PZXKAXZ54UUHU72S","created_at":"2026-05-18T12:32:46.962924+00:00"},{"alias_kind":"pith_short_8","alias_value":"PZXKAXZ5","created_at":"2026-05-18T12:32:46.962924+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/PZXKAXZ54UUHU72S3UTNC6WQUN","json":"https://pith.science/pith/PZXKAXZ54UUHU72S3UTNC6WQUN.json","graph_json":"https://pith.science/api/pith-number/PZXKAXZ54UUHU72S3UTNC6WQUN/graph.json","events_json":"https://pith.science/api/pith-number/PZXKAXZ54UUHU72S3UTNC6WQUN/events.json","paper":"https://pith.science/paper/PZXKAXZ5"},"agent_actions":{"view_html":"https://pith.science/pith/PZXKAXZ54UUHU72S3UTNC6WQUN","download_json":"https://pith.science/pith/PZXKAXZ54UUHU72S3UTNC6WQUN.json","view_paper":"https://pith.science/paper/PZXKAXZ5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1801.01047&json=true","fetch_graph":"https://pith.science/api/pith-number/PZXKAXZ54UUHU72S3UTNC6WQUN/graph.json","fetch_events":"https://pith.science/api/pith-number/PZXKAXZ54UUHU72S3UTNC6WQUN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PZXKAXZ54UUHU72S3UTNC6WQUN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PZXKAXZ54UUHU72S3UTNC6WQUN/action/storage_attestation","attest_author":"https://pith.science/pith/PZXKAXZ54UUHU72S3UTNC6WQUN/action/author_attestation","sign_citation":"https://pith.science/pith/PZXKAXZ54UUHU72S3UTNC6WQUN/action/citation_signature","submit_replication":"https://pith.science/pith/PZXKAXZ54UUHU72S3UTNC6WQUN/action/replication_record"}},"created_at":"2026-05-18T00:26:07.240366+00:00","updated_at":"2026-05-18T00:26:07.240366+00:00"}