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We develop a \\emph{palette framework} for this density. For every family $\\mathcal F$ of $k$-graphs, we prove that $\\pi_{k-2}(\\mathcal F)$ equals the corresponding palette Tur\\'an density. We further establish palette classification tools for the existence of $k$-graphs satisfying prescribed palette colorability constr"},"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":true,"formal_links_present":true},"canonical_record":{"source":{"id":"2605.15105","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-14T17:23:04Z","cross_cats_sorted":[],"title_canon_sha256":"d7efcb6681589ed4655b3c175f2aec4379df8cb87e7827bb456f500ead80b6a9","abstract_canon_sha256":"e7ef263ca68272e1c715e7b6c3328fa0be845b7c39f3369a9f82221595dccc0c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.2","canonical_sha256":"96b0861d0268b44750481e870f16ed7e66f67f2541bc6a94034cac7b8d8f706d","last_reissued_at":"2026-05-17T21:57:19.127486Z","signature_status":"unsigned_v0","first_computed_at":"2026-05-17T21:40:25.795609Z"},"graph_snapshot":{"paper":{"title":"Uniform Tur\\'an densities of $k$-uniform hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"For any family of k-graphs, the (k-2)-uniform Turán density equals the palette Turán density.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Guanghui Wang, Guowei Sun, Hao Lin, Wenling Zhou","submitted_at":"2026-05-14T17:23:04Z","abstract_excerpt":"For $k\\ge 3$, the $(k-2)$-uniform Tur\\'an density $\\pi_{k-2}(F)$ of a $k$-graph $F$ is the supremum of $d$ for which there are arbitrarily large $F$-free $k$-graphs that are uniformly $d$-dense with respect to the $k$-vertex cliques of every $(k-2)$-graph on the same vertex set. We develop a \\emph{palette framework} for this density. For every family $\\mathcal F$ of $k$-graphs, we prove that $\\pi_{k-2}(\\mathcal F)$ equals the corresponding palette Tur\\'an density. We further establish palette classification tools for the existence of $k$-graphs satisfying prescribed palette colorability constr"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For every family F of k-graphs, we prove that π_{k-2}(F) equals the corresponding palette Turán density. ... we establish the following values [list of six expressions] as (k-2)-uniform Turán densities of single k-graphs. Finally ... there exist k-graphs F1,F2 such that π_{k-2}({F1,F2}) < min{π_{k-2}(F1),π_{k-2}(F2)}.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The palette classification tools correctly characterize the existence of k-graphs satisfying prescribed palette colorability constraints, and these tools apply without hidden restrictions on the underlying vertex sets or color palettes.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A new palette framework reduces (k-2)-uniform Turán densities of k-graphs to palette-homomorphism problems and yields exact values including (r-1)/r, (r-1)^2/r^2, and (k-1)^k/k^k for various k and r.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For any family of k-graphs, the (k-2)-uniform Turán density equals the palette Turán density.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"865fd8208137392df14498b20f99aa7fbd3e56a8aa39f612ffe40ed393c5fe06"},"source":{"id":"2605.15105","kind":"arxiv","version":1},"verdict":{"id":"3c592261-d041-4b4d-9371-6859f9c082fc","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T03:15:45.126868Z","strongest_claim":"For every family F of k-graphs, we prove that π_{k-2}(F) equals the corresponding palette Turán density. ... we establish the following values [list of six expressions] as (k-2)-uniform Turán densities of single k-graphs. Finally ... there exist k-graphs F1,F2 such that π_{k-2}({F1,F2}) < min{π_{k-2}(F1),π_{k-2}(F2)}.","one_line_summary":"A new palette framework reduces (k-2)-uniform Turán densities of k-graphs to palette-homomorphism problems and yields exact values including (r-1)/r, (r-1)^2/r^2, and (k-1)^k/k^k for various k and r.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The palette classification tools correctly characterize the existence of k-graphs satisfying prescribed palette colorability constraints, and these tools apply without hidden restrictions on the underlying vertex sets or color palettes.","pith_extraction_headline":"For any family of k-graphs, the (k-2)-uniform Turán density equals the palette Turán density."},"references":{"count":31,"sample":[{"doi":"","year":2002,"title":"J. Balogh. The Turán density of triple systems is not principal.J. Combin. Theory Ser. 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