{"paper":{"title":"Lagrangians are attained as uniform Tur\\'an densities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bjarne Sch\\\"ulke, Dylan King, Marcelo Sales","submitted_at":"2024-12-10T08:27:49Z","abstract_excerpt":"The study of uniform Tur\\'an densities was initiated in the 1980s by Erd\\H{o}s and S\\'os. Given a $3$-graph $F$, the uniform Tur\\'an density of $F$, $\\pi_{\\therefore}(F)$, is defined as the infimum $d\\in[0,1]$ such that every $3$-graph $H$ in which every linearly sized $S\\subseteq V(H)$ induces at least $(d+o(1))\\binom{\\vert S\\vert}{3}$ edges must contain a copy of $F$. Disproving Erd\\H{o}s's famous jumping conjecture, Frankl and R\\\"odl showed that the set of Tur\\'an densities is not well-ordered. We prove an analogous result for the uniform Tur\\'an density, namely that the set $\\Pi^{(3)}_{\\th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.07297","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/2412.07297/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"}