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Balogh, Bohman, Bollob\\'as, and Zhao previously obtained bounds for these deficits with logarithmic gaps near both ends of the sequence $B_i^{(r)}$, namely, when $i=O(1)$ or $i=r-O(1)$. We close these gaps by showing that, as $r\\to\\infty$, for every fixed integer $a\\ge1$, $q_{r,a+1}=\\Theta_a(r^{-a})$, and for every fixed integer $b\\ge2$, $"},"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":"2606.12133","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-10T14:28:25Z","cross_cats_sorted":[],"title_canon_sha256":"610b9910a5bf5b48daf0431f4d06f85ad9bf5381d9f3c80eef11e301e74d92f2","abstract_canon_sha256":"5449e0f4ad633b6073b0154061cff029ba132989029db2bce981af3f304e32f5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-11T01:10:50.089203Z","signature_b64":"WvllWrGjZA1XkH31+n9xFq2IwS6MmCdQyZ236p4pNDoRrHEV6R9sk8dTeET74lIAte2ldni/9hth7E9Nkgx7BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dd60289ceadb30721d3ee932b40db4371061d3827bf123554ea89d1049d9318e","last_reissued_at":"2026-06-11T01:10:50.088412Z","signature_status":"signed_v1","first_computed_at":"2026-06-11T01:10:50.088412Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On a hypergraph Tur\\'an problem of Balogh-Bohman-Bollob\\'as-Zhao","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Heng Li, Jie Ma, Tianhen Wang, Tianming Zhu, Yixiao Zhang","submitted_at":"2026-06-10T14:28:25Z","abstract_excerpt":"Let $S$ and $T$ be disjoint sets with $|S|=i$ and $|T|=r-1$ for $2\\le i\\le r-1$, and let $B_i^{(r)}$ be the $r$-graph on $S\\cup T$ whose edges are the $r$-subsets containing $S$ or $T$. We study the deficit $q_{r,i}:=1-\\pi(B_i^{(r)})$ in its Tur\\'an density. Balogh, Bohman, Bollob\\'as, and Zhao previously obtained bounds for these deficits with logarithmic gaps near both ends of the sequence $B_i^{(r)}$, namely, when $i=O(1)$ or $i=r-O(1)$. We close these gaps by showing that, as $r\\to\\infty$, for every fixed integer $a\\ge1$, $q_{r,a+1}=\\Theta_a(r^{-a})$, and for every fixed integer $b\\ge2$, $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.12133","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/2606.12133/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":"2606.12133","created_at":"2026-06-11T01:10:50.088531+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.12133v1","created_at":"2026-06-11T01:10:50.088531+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.12133","created_at":"2026-06-11T01:10:50.088531+00:00"},{"alias_kind":"pith_short_12","alias_value":"3VQCRHHK3MYH","created_at":"2026-06-11T01:10:50.088531+00:00"},{"alias_kind":"pith_short_16","alias_value":"3VQCRHHK3MYHEHJ6","created_at":"2026-06-11T01:10:50.088531+00:00"},{"alias_kind":"pith_short_8","alias_value":"3VQCRHHK","created_at":"2026-06-11T01:10:50.088531+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/3VQCRHHK3MYHEHJ65EZLIDNUG4","json":"https://pith.science/pith/3VQCRHHK3MYHEHJ65EZLIDNUG4.json","graph_json":"https://pith.science/api/pith-number/3VQCRHHK3MYHEHJ65EZLIDNUG4/graph.json","events_json":"https://pith.science/api/pith-number/3VQCRHHK3MYHEHJ65EZLIDNUG4/events.json","paper":"https://pith.science/paper/3VQCRHHK"},"agent_actions":{"view_html":"https://pith.science/pith/3VQCRHHK3MYHEHJ65EZLIDNUG4","download_json":"https://pith.science/pith/3VQCRHHK3MYHEHJ65EZLIDNUG4.json","view_paper":"https://pith.science/paper/3VQCRHHK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.12133&json=true","fetch_graph":"https://pith.science/api/pith-number/3VQCRHHK3MYHEHJ65EZLIDNUG4/graph.json","fetch_events":"https://pith.science/api/pith-number/3VQCRHHK3MYHEHJ65EZLIDNUG4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3VQCRHHK3MYHEHJ65EZLIDNUG4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3VQCRHHK3MYHEHJ65EZLIDNUG4/action/storage_attestation","attest_author":"https://pith.science/pith/3VQCRHHK3MYHEHJ65EZLIDNUG4/action/author_attestation","sign_citation":"https://pith.science/pith/3VQCRHHK3MYHEHJ65EZLIDNUG4/action/citation_signature","submit_replication":"https://pith.science/pith/3VQCRHHK3MYHEHJ65EZLIDNUG4/action/replication_record"}},"created_at":"2026-06-11T01:10:50.088531+00:00","updated_at":"2026-06-11T01:10:50.088531+00:00"}