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Let \\[ \\mu\\left( G\\right) :=\\max {\\textstyle\\sum\\limits_{\\left\\{ i_{1},\\ldots,i_{r}\\right\\} \\in E}} x_{i_{1}}\\cdots x_{i_{r}}, \\] where the maximum is taken over all nonnegative $x_{1},\\ldots,x_{n}$ with $x_{1}+\\cdots+x_{n}=1.$\n  Let $t\\geq r-1$ be the unique real number such that $\\left\\vert E\\right\\vert =\\binom{t}{r}$. It is shown that if $r\\leq5$ or $t\\geq4\\left( r-1\\right) \\left( r-2\\right) $, then \\[ \\mu\\left( G\\right) \\leq t^{-r}\\binom{t}{r}% \\] with equality holding if and o"},"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":"1802.10075","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-02-27T18:56:55Z","cross_cats_sorted":[],"title_canon_sha256":"80c55fb92f7bb03789c279b876b68a76b66e051fd651877486380711f5bbfa7f","abstract_canon_sha256":"8211cc0363d747b21d1cdf59cf7a5a94b380e00255cd7a7e5c29ae49e4e1bbb5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:21:04.222057Z","signature_b64":"+Ah2xRrCOl7XDii9Yge/GU8iKRHES16cIN/P5vNMmHVH9+f/c2RWGlrujqA7k4Sr85CrS8e7h+ezyNHc0d66Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a2048d949da0b4d221506aabf36010b8ae33bdb90592592b90515c009533155d","last_reissued_at":"2026-05-18T00:21:04.221625Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:21:04.221625Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Symmetric functions and the principal case of the Frankl-F\\\"uredi conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"V. Nikiforov","submitted_at":"2018-02-27T18:56:55Z","abstract_excerpt":"Let $r\\geq3$ and $G$ be an $r$-uniform hypergraph with vertex set $\\left\\{ 1,\\ldots,n\\right\\} $ and edge set $E$. Let \\[ \\mu\\left( G\\right) :=\\max {\\textstyle\\sum\\limits_{\\left\\{ i_{1},\\ldots,i_{r}\\right\\} \\in E}} x_{i_{1}}\\cdots x_{i_{r}}, \\] where the maximum is taken over all nonnegative $x_{1},\\ldots,x_{n}$ with $x_{1}+\\cdots+x_{n}=1.$\n  Let $t\\geq r-1$ be the unique real number such that $\\left\\vert E\\right\\vert =\\binom{t}{r}$. It is shown that if $r\\leq5$ or $t\\geq4\\left( r-1\\right) \\left( r-2\\right) $, then \\[ \\mu\\left( G\\right) \\leq t^{-r}\\binom{t}{r}% \\] with equality holding if and o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.10075","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"1802.10075","created_at":"2026-05-18T00:21:04.221689+00:00"},{"alias_kind":"arxiv_version","alias_value":"1802.10075v3","created_at":"2026-05-18T00:21:04.221689+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.10075","created_at":"2026-05-18T00:21:04.221689+00:00"},{"alias_kind":"pith_short_12","alias_value":"UICI3FE5UC2N","created_at":"2026-05-18T12:32:56.356000+00:00"},{"alias_kind":"pith_short_16","alias_value":"UICI3FE5UC2NEIKQ","created_at":"2026-05-18T12:32:56.356000+00:00"},{"alias_kind":"pith_short_8","alias_value":"UICI3FE5","created_at":"2026-05-18T12:32:56.356000+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/UICI3FE5UC2NEIKQNKV7GYAQXC","json":"https://pith.science/pith/UICI3FE5UC2NEIKQNKV7GYAQXC.json","graph_json":"https://pith.science/api/pith-number/UICI3FE5UC2NEIKQNKV7GYAQXC/graph.json","events_json":"https://pith.science/api/pith-number/UICI3FE5UC2NEIKQNKV7GYAQXC/events.json","paper":"https://pith.science/paper/UICI3FE5"},"agent_actions":{"view_html":"https://pith.science/pith/UICI3FE5UC2NEIKQNKV7GYAQXC","download_json":"https://pith.science/pith/UICI3FE5UC2NEIKQNKV7GYAQXC.json","view_paper":"https://pith.science/paper/UICI3FE5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1802.10075&json=true","fetch_graph":"https://pith.science/api/pith-number/UICI3FE5UC2NEIKQNKV7GYAQXC/graph.json","fetch_events":"https://pith.science/api/pith-number/UICI3FE5UC2NEIKQNKV7GYAQXC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UICI3FE5UC2NEIKQNKV7GYAQXC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UICI3FE5UC2NEIKQNKV7GYAQXC/action/storage_attestation","attest_author":"https://pith.science/pith/UICI3FE5UC2NEIKQNKV7GYAQXC/action/author_attestation","sign_citation":"https://pith.science/pith/UICI3FE5UC2NEIKQNKV7GYAQXC/action/citation_signature","submit_replication":"https://pith.science/pith/UICI3FE5UC2NEIKQNKV7GYAQXC/action/replication_record"}},"created_at":"2026-05-18T00:21:04.221689+00:00","updated_at":"2026-05-18T00:21:04.221689+00:00"}