{"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. A, 100(1):176–180, 2002","work_id":"da375963-25ec-4ba1-9c00-e5d90dedfa78","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1969,"title":"Colloq., Balatonfüred, 1969), volume 4 ofColloq","work_id":"00fd6110-fabc-40a4-90f5-50025ce220b0","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2023,"title":"M. Bucić, J. W. Cooper, D. Král’, S. Mohr, and D. Munhá Correia. Uniform Turán density of cycles.Trans. Amer. Math. Soc., 376(7):4765–4809, 2023","work_id":"f31abd78-e169-4989-93e9-2bb4f468ebda","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2026,"title":"A. Y. Chen and B. Schülke. Beyond the broken tetrahedron.Combin. Probab. Comput., 35(1):59–70, 2026","work_id":"db84fc52-5505-419c-a0a3-46465d9f56cf","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2009,"title":"O. Cooley, N. Fountoulakis, D. Kühn, and D. Osthus. Embeddings and Ramsey numbers of sparsek-uniform hypergraphs.Combinatorica, 29(3):263–297, 2009","work_id":"43ae44d9-5ace-4589-9f2a-72046170804f","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":31,"snapshot_sha256":"7ca160205783893f4c00c42c4a11a03e4d080fd6562a41e9288792e408f9a9c4","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"6a6460af3242882b4d38e85477176a4c72cd4b3e5d522e7eca41037974deb8c4"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}