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Let $c(n,F)$ be the minimum number of copies obtained by adding one edge to an extremal $F$-free graph. Mubayi's supersaturation conjecture predicts, under a stability hypothesis, that $h_F(n,q)\\ge q\\,c(n,F)$. Ma and Yuan recently constructed stable graph counterexamples for every fixed $q\\ge4$; they asked whether the one-edge equality $h_F(n,1)=c(n,F)$ might still hold for every g"},"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":"2606.09518","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-08T14:08:27Z","cross_cats_sorted":[],"title_canon_sha256":"16d7f40441e03287636f653121b118c569b5fc3974220cf0be1a7d083322f159","abstract_canon_sha256":"005c147ba311b3e6631cdd06560003f1d2e367a74d1cc0466423318ad59537f7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-09T02:08:53.039030Z","signature_b64":"yFDV+MQSbjrMAMQoqf9YVzauLuX+M2tFy14gag53o++EwkSozAmzDzDETQd+FfgzIRkTRYaefHwvPDySekvYBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3f1658fde266f6c4a66981cc30239b14e80376dc68a115fee96eb35aa0e11aef","last_reissued_at":"2026-06-09T02:08:53.038181Z","signature_status":"signed_v1","first_computed_at":"2026-06-09T02:08:53.038181Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Strong counterexamples to a supersaturation question of Ma-Yuan","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Long-tu Yuan, Wanfang Chen","submitted_at":"2026-06-08T14:08:27Z","abstract_excerpt":"For a graph $F$, let $h_F(n,q)$ be the minimum number of copies of $F$ in an $n$-vertex graph with $\\mathrm{ex}(n,F)+q$ edges, where $\\mathrm{ex}(n,F)$ is the maximum number of edges in an $n$-vertex $F$-free graph. Let $c(n,F)$ be the minimum number of copies obtained by adding one edge to an extremal $F$-free graph. Mubayi's supersaturation conjecture predicts, under a stability hypothesis, that $h_F(n,q)\\ge q\\,c(n,F)$. Ma and Yuan recently constructed stable graph counterexamples for every fixed $q\\ge4$; they asked whether the one-edge equality $h_F(n,1)=c(n,F)$ might still hold for every g"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.09518","kind":"arxiv","version":1},"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/2606.09518/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":"2606.09518","created_at":"2026-06-09T02:08:53.038334+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.09518v1","created_at":"2026-06-09T02:08:53.038334+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.09518","created_at":"2026-06-09T02:08:53.038334+00:00"},{"alias_kind":"pith_short_12","alias_value":"H4LFR7PCM33M","created_at":"2026-06-09T02:08:53.038334+00:00"},{"alias_kind":"pith_short_16","alias_value":"H4LFR7PCM33MJJTJ","created_at":"2026-06-09T02:08:53.038334+00:00"},{"alias_kind":"pith_short_8","alias_value":"H4LFR7PC","created_at":"2026-06-09T02:08:53.038334+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/H4LFR7PCM33MJJTJQHGDAI43CT","json":"https://pith.science/pith/H4LFR7PCM33MJJTJQHGDAI43CT.json","graph_json":"https://pith.science/api/pith-number/H4LFR7PCM33MJJTJQHGDAI43CT/graph.json","events_json":"https://pith.science/api/pith-number/H4LFR7PCM33MJJTJQHGDAI43CT/events.json","paper":"https://pith.science/paper/H4LFR7PC"},"agent_actions":{"view_html":"https://pith.science/pith/H4LFR7PCM33MJJTJQHGDAI43CT","download_json":"https://pith.science/pith/H4LFR7PCM33MJJTJQHGDAI43CT.json","view_paper":"https://pith.science/paper/H4LFR7PC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.09518&json=true","fetch_graph":"https://pith.science/api/pith-number/H4LFR7PCM33MJJTJQHGDAI43CT/graph.json","fetch_events":"https://pith.science/api/pith-number/H4LFR7PCM33MJJTJQHGDAI43CT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/H4LFR7PCM33MJJTJQHGDAI43CT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/H4LFR7PCM33MJJTJQHGDAI43CT/action/storage_attestation","attest_author":"https://pith.science/pith/H4LFR7PCM33MJJTJQHGDAI43CT/action/author_attestation","sign_citation":"https://pith.science/pith/H4LFR7PCM33MJJTJQHGDAI43CT/action/citation_signature","submit_replication":"https://pith.science/pith/H4LFR7PCM33MJJTJQHGDAI43CT/action/replication_record"}},"created_at":"2026-06-09T02:08:53.038334+00:00","updated_at":"2026-06-09T02:08:53.038334+00:00"}