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In particular, it implies that for large $n$, the unique extremal triangle-free construction on $n$ vertices is the balanced complete $r$-partite $r$-graph. The latter result answers a question b"},"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":"2501.19229","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-01-31T15:41:08Z","cross_cats_sorted":[],"title_canon_sha256":"8fb096eb81426b4110f844f60a5151398f9baecf8a0f5733bacc9231541df24c","abstract_canon_sha256":"5a7e930be766856bd9a8e1db0ba2d4c6ca74fcfa937d02f243e7caba093f6c03"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:07:58.027889Z","signature_b64":"V5M0AEDoWNug3zt6xWcz6uliDjR1fSDFE0r+KwxseBMDhJYqs14FZU4mxpXeOzyX8Qv/cZDE12GJnrCoBd0XBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"decb8dc8e079fbf582557238105cb4773098df24c7b73044494c84ff84f90ac8","last_reissued_at":"2026-07-05T10:07:58.027435Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:07:58.027435Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On a hypergraph Mantel theorem","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Xizhi Liu","submitted_at":"2025-01-31T15:41:08Z","abstract_excerpt":"An $r$-graph is a triangle if there exists a positive integer $i \\le \\lceil r/2 \\rceil$ such that it is isomorphic to the following $r$-graph with three edges: \\begin{align*}\n  \\left\\{\\{1, \\ldots, r\\},~\\{1, \\ldots, i, r+1, \\ldots, 2r-i\\},~\\{i+1, \\ldots, r, r+1, 2r-i+1, \\ldots,2r-1\\}\\right\\}. \\end{align*} We prove an Andr{\\'a}sfai--Erd\\H{o}s--S\\'{o}s-type stability theorem for triangle-free $r$-graphs. In particular, it implies that for large $n$, the unique extremal triangle-free construction on $n$ vertices is the balanced complete $r$-partite $r$-graph. The latter result answers a question b"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.19229","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/2501.19229/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":"2501.19229","created_at":"2026-07-05T10:07:58.027492+00:00"},{"alias_kind":"arxiv_version","alias_value":"2501.19229v1","created_at":"2026-07-05T10:07:58.027492+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2501.19229","created_at":"2026-07-05T10:07:58.027492+00:00"},{"alias_kind":"pith_short_12","alias_value":"33FY3SHAPH57","created_at":"2026-07-05T10:07:58.027492+00:00"},{"alias_kind":"pith_short_16","alias_value":"33FY3SHAPH57LASV","created_at":"2026-07-05T10:07:58.027492+00:00"},{"alias_kind":"pith_short_8","alias_value":"33FY3SHA","created_at":"2026-07-05T10:07:58.027492+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.22828","citing_title":"Tur\\'an numbers of $4$-uniform tight even cycles minus one edge","ref_index":21,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/33FY3SHAPH57LASVOI4BAXFUO4","json":"https://pith.science/pith/33FY3SHAPH57LASVOI4BAXFUO4.json","graph_json":"https://pith.science/api/pith-number/33FY3SHAPH57LASVOI4BAXFUO4/graph.json","events_json":"https://pith.science/api/pith-number/33FY3SHAPH57LASVOI4BAXFUO4/events.json","paper":"https://pith.science/paper/33FY3SHA"},"agent_actions":{"view_html":"https://pith.science/pith/33FY3SHAPH57LASVOI4BAXFUO4","download_json":"https://pith.science/pith/33FY3SHAPH57LASVOI4BAXFUO4.json","view_paper":"https://pith.science/paper/33FY3SHA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2501.19229&json=true","fetch_graph":"https://pith.science/api/pith-number/33FY3SHAPH57LASVOI4BAXFUO4/graph.json","fetch_events":"https://pith.science/api/pith-number/33FY3SHAPH57LASVOI4BAXFUO4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/33FY3SHAPH57LASVOI4BAXFUO4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/33FY3SHAPH57LASVOI4BAXFUO4/action/storage_attestation","attest_author":"https://pith.science/pith/33FY3SHAPH57LASVOI4BAXFUO4/action/author_attestation","sign_citation":"https://pith.science/pith/33FY3SHAPH57LASVOI4BAXFUO4/action/citation_signature","submit_replication":"https://pith.science/pith/33FY3SHAPH57LASVOI4BAXFUO4/action/replication_record"}},"created_at":"2026-07-05T10:07:58.027492+00:00","updated_at":"2026-07-05T10:07:58.027492+00:00"}