{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:MDIVNUVNCL6QL2EG5XE3HBU7SJ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"d414b0b1069f4e94a5c316c1b36bcd688b20ad47fe21dae8bf77aab7e3fa3bf7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2022-01-11T14:59:59Z","title_canon_sha256":"2a9897ce3b8df9b307805cbcb70f55fc156a2133fca9d40a1ac366c9147cb775"},"schema_version":"1.0","source":{"id":"2201.03983","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2201.03983","created_at":"2026-07-05T03:48:11Z"},{"alias_kind":"arxiv_version","alias_value":"2201.03983v2","created_at":"2026-07-05T03:48:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2201.03983","created_at":"2026-07-05T03:48:11Z"},{"alias_kind":"pith_short_12","alias_value":"MDIVNUVNCL6Q","created_at":"2026-07-05T03:48:11Z"},{"alias_kind":"pith_short_16","alias_value":"MDIVNUVNCL6QL2EG","created_at":"2026-07-05T03:48:11Z"},{"alias_kind":"pith_short_8","alias_value":"MDIVNUVN","created_at":"2026-07-05T03:48:11Z"}],"graph_snapshots":[{"event_id":"sha256:4e27e48dbaf3bf18be5517c0b6e5773fbf70f8da476df7ced124cc478ec8bf24","target":"graph","created_at":"2026-07-05T03:48:11Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2201.03983/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"Fuhong Ma, Jialin He, Jie Ma, Xinyang Ye","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2022-01-11T14:59:59Z","title":"The minimum number of clique-saturating edges"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2201.03983","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:32e8e56c536a9a7b2c0c32794397304a59210f5ef016e2a983a75da825ca5169","target":"record","created_at":"2026-07-05T03:48:11Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"d414b0b1069f4e94a5c316c1b36bcd688b20ad47fe21dae8bf77aab7e3fa3bf7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2022-01-11T14:59:59Z","title_canon_sha256":"2a9897ce3b8df9b307805cbcb70f55fc156a2133fca9d40a1ac366c9147cb775"},"schema_version":"1.0","source":{"id":"2201.03983","kind":"arxiv","version":2}},"canonical_sha256":"60d156d2ad12fd05e886edc9b3869f9265ce3350a8827e36987578f0f3e01c7b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"60d156d2ad12fd05e886edc9b3869f9265ce3350a8827e36987578f0f3e01c7b","first_computed_at":"2026-07-05T03:48:11.055338Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:48:11.055338Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6R9nZ4lKrPhVvxW/RorRYerEDBufZ+KyT6CePSDCqhizJ4aF/GeOhwisr+nBuHvIM/6XMlqpb0ibd85k5drWDw==","signature_status":"signed_v1","signed_at":"2026-07-05T03:48:11.055814Z","signed_message":"canonical_sha256_bytes"},"source_id":"2201.03983","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:32e8e56c536a9a7b2c0c32794397304a59210f5ef016e2a983a75da825ca5169","sha256:4e27e48dbaf3bf18be5517c0b6e5773fbf70f8da476df7ced124cc478ec8bf24"],"state_sha256":"5cefbdbcf8c00cb5c48287eaea32c1dae05efafef690640011b22d1a61788212"}