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We confirm this conjecture. We also determine exactly the maximum number of cliques in an $n$-vertex $K_{5}^{3}$-free $3$-uniform hypergraph for all sufficiently large $n$, thereby verifying the corresponding case of a conjecture of Frankl, Gryaznov, and Talebanfard.\n  The main ingredients are Tur\\'{a}n-type theorems for vertex-co"},"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":"2606.02210","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-01T13:10:40Z","cross_cats_sorted":[],"title_canon_sha256":"ba9dbc91f1717348ea0b3ba360b26fa8d5e658386117ebacd15060b5dcd97fc4","abstract_canon_sha256":"1806eeaf2e63980006c6f5d96fdb13a2819e70eb76040f2a32d915167b2e4d4c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T03:04:53.179095Z","signature_b64":"+dvYLnNWX8Dj4zB+nSW6DCEtWz6mfU5dDnqGn4z1dAjArkroPci2yFQecpIqHU8gxi6w5YyvrNNOP6Q69bJkAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"01e713636cdd2de595dcf8ff17a38fb630cbb12849f340ca9b2b012a1ba4a83b","last_reissued_at":"2026-06-02T03:04:53.178760Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T03:04:53.178760Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Vertex-colored Tur\\'{a}n theorems with applications in extremal hypergraph problems","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jianfeng Hou, Jinghua Deng, Wanfang Chen, Xizhi Liu, Yixiao Zhang","submitted_at":"2026-06-01T13:10:40Z","abstract_excerpt":"Balogh, Clemen, and Lidick\\'{y} proved that the $\\ell_{2}$-norm Tur\\'{a}n problem for $K_{5}^{3}$ is asymptotically solved by the balanced bipartite construction, and they further conjectured that this construction is uniquely extremal for all sufficiently large $n$. We confirm this conjecture. We also determine exactly the maximum number of cliques in an $n$-vertex $K_{5}^{3}$-free $3$-uniform hypergraph for all sufficiently large $n$, thereby verifying the corresponding case of a conjecture of Frankl, Gryaznov, and Talebanfard.\n  The main ingredients are Tur\\'{a}n-type theorems for vertex-co"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.02210","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/2606.02210/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":"2606.02210","created_at":"2026-06-02T03:04:53.178816+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.02210v1","created_at":"2026-06-02T03:04:53.178816+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.02210","created_at":"2026-06-02T03:04:53.178816+00:00"},{"alias_kind":"pith_short_12","alias_value":"AHTRGY3M3UW6","created_at":"2026-06-02T03:04:53.178816+00:00"},{"alias_kind":"pith_short_16","alias_value":"AHTRGY3M3UW6LFO4","created_at":"2026-06-02T03:04:53.178816+00:00"},{"alias_kind":"pith_short_8","alias_value":"AHTRGY3M","created_at":"2026-06-02T03:04:53.178816+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/AHTRGY3M3UW6LFO47D7RPI4PWY","json":"https://pith.science/pith/AHTRGY3M3UW6LFO47D7RPI4PWY.json","graph_json":"https://pith.science/api/pith-number/AHTRGY3M3UW6LFO47D7RPI4PWY/graph.json","events_json":"https://pith.science/api/pith-number/AHTRGY3M3UW6LFO47D7RPI4PWY/events.json","paper":"https://pith.science/paper/AHTRGY3M"},"agent_actions":{"view_html":"https://pith.science/pith/AHTRGY3M3UW6LFO47D7RPI4PWY","download_json":"https://pith.science/pith/AHTRGY3M3UW6LFO47D7RPI4PWY.json","view_paper":"https://pith.science/paper/AHTRGY3M","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.02210&json=true","fetch_graph":"https://pith.science/api/pith-number/AHTRGY3M3UW6LFO47D7RPI4PWY/graph.json","fetch_events":"https://pith.science/api/pith-number/AHTRGY3M3UW6LFO47D7RPI4PWY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AHTRGY3M3UW6LFO47D7RPI4PWY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AHTRGY3M3UW6LFO47D7RPI4PWY/action/storage_attestation","attest_author":"https://pith.science/pith/AHTRGY3M3UW6LFO47D7RPI4PWY/action/author_attestation","sign_citation":"https://pith.science/pith/AHTRGY3M3UW6LFO47D7RPI4PWY/action/citation_signature","submit_replication":"https://pith.science/pith/AHTRGY3M3UW6LFO47D7RPI4PWY/action/replication_record"}},"created_at":"2026-06-02T03:04:53.178816+00:00","updated_at":"2026-06-02T03:04:53.178816+00:00"}