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Erd\\H{o}s and Tuza conjectured that $f_4(n,\\lfloor n^2/4\\rfloor+1)=\\left(1 + o(1)\\right)\\frac{n^2}{16}.$ Balogh and Liu disproved this by showing $f_4(n,\\lfloor n^2/4\\rfloor+1)=(1+o(1))\\frac{2n^2}{33}$. They believed that a natural generalization of their construction for $K_p$-free graph should also be optimal and made a conjecture that $f"},"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":"2201.03983","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2022-01-11T14:59:59Z","cross_cats_sorted":[],"title_canon_sha256":"2a9897ce3b8df9b307805cbcb70f55fc156a2133fca9d40a1ac366c9147cb775","abstract_canon_sha256":"d414b0b1069f4e94a5c316c1b36bcd688b20ad47fe21dae8bf77aab7e3fa3bf7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:48:11.055814Z","signature_b64":"6R9nZ4lKrPhVvxW/RorRYerEDBufZ+KyT6CePSDCqhizJ4aF/GeOhwisr+nBuHvIM/6XMlqpb0ibd85k5drWDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"60d156d2ad12fd05e886edc9b3869f9265ce3350a8827e36987578f0f3e01c7b","last_reissued_at":"2026-07-05T03:48:11.055338Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:48:11.055338Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The minimum number of clique-saturating edges","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Fuhong Ma, Jialin He, Jie Ma, Xinyang Ye","submitted_at":"2022-01-11T14:59:59Z","abstract_excerpt":"Let $G$ be a $K_p$-free graph. We say $e$ is a $K_p$-saturating edge of $G$ if $e\\notin E(G)$ and $G+e$ contains a copy of $K_p$. Denote by $f_p(n, e)$ the minimum number of $K_p$-saturating edges that an $n$-vertex $K_p$-free graph with $e$ edges can have. Erd\\H{o}s and Tuza conjectured that $f_4(n,\\lfloor n^2/4\\rfloor+1)=\\left(1 + o(1)\\right)\\frac{n^2}{16}.$ Balogh and Liu disproved this by showing $f_4(n,\\lfloor n^2/4\\rfloor+1)=(1+o(1))\\frac{2n^2}{33}$. They believed that a natural generalization of their construction for $K_p$-free graph should also be optimal and made a conjecture that $f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2201.03983","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2201.03983/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2201.03983","created_at":"2026-07-05T03:48:11.055410+00:00"},{"alias_kind":"arxiv_version","alias_value":"2201.03983v2","created_at":"2026-07-05T03:48:11.055410+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2201.03983","created_at":"2026-07-05T03:48:11.055410+00:00"},{"alias_kind":"pith_short_12","alias_value":"MDIVNUVNCL6Q","created_at":"2026-07-05T03:48:11.055410+00:00"},{"alias_kind":"pith_short_16","alias_value":"MDIVNUVNCL6QL2EG","created_at":"2026-07-05T03:48:11.055410+00:00"},{"alias_kind":"pith_short_8","alias_value":"MDIVNUVN","created_at":"2026-07-05T03:48:11.055410+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/MDIVNUVNCL6QL2EG5XE3HBU7SJ","json":"https://pith.science/pith/MDIVNUVNCL6QL2EG5XE3HBU7SJ.json","graph_json":"https://pith.science/api/pith-number/MDIVNUVNCL6QL2EG5XE3HBU7SJ/graph.json","events_json":"https://pith.science/api/pith-number/MDIVNUVNCL6QL2EG5XE3HBU7SJ/events.json","paper":"https://pith.science/paper/MDIVNUVN"},"agent_actions":{"view_html":"https://pith.science/pith/MDIVNUVNCL6QL2EG5XE3HBU7SJ","download_json":"https://pith.science/pith/MDIVNUVNCL6QL2EG5XE3HBU7SJ.json","view_paper":"https://pith.science/paper/MDIVNUVN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2201.03983&json=true","fetch_graph":"https://pith.science/api/pith-number/MDIVNUVNCL6QL2EG5XE3HBU7SJ/graph.json","fetch_events":"https://pith.science/api/pith-number/MDIVNUVNCL6QL2EG5XE3HBU7SJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MDIVNUVNCL6QL2EG5XE3HBU7SJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MDIVNUVNCL6QL2EG5XE3HBU7SJ/action/storage_attestation","attest_author":"https://pith.science/pith/MDIVNUVNCL6QL2EG5XE3HBU7SJ/action/author_attestation","sign_citation":"https://pith.science/pith/MDIVNUVNCL6QL2EG5XE3HBU7SJ/action/citation_signature","submit_replication":"https://pith.science/pith/MDIVNUVNCL6QL2EG5XE3HBU7SJ/action/replication_record"}},"created_at":"2026-07-05T03:48:11.055410+00:00","updated_at":"2026-07-05T03:48:11.055410+00:00"}