{"paper":{"title":"Stability and Tur\\'an numbers of a class of hypergraphs via Lagrangians","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Axel Brandt, David Irwin, Tao Jiang","submitted_at":"2015-10-12T20:59:21Z","abstract_excerpt":"Given a family of $r$-uniform hypergraphs ${\\cal F}$ (or $r$-graphs for brevity), the Tur\\'an number $ex(n,{\\cal F})$ of ${\\cal F}$ is the maximum number of edges in an $r$-graph on $n$ vertices that does not contain any member of ${\\cal F}$. A pair $\\{u,v\\}$ is covered in a hypergraph $G$ if some edge of $G$ contains $\\{u,v\\}$. Given an $r$-graph $F$ and a positive integer $p\\geq n(F)$, let $H^F_p$ denote the $r$-graph obtained as follows. Label the vertices of $F$ as $v_1,\\ldots, v_{n(F)}$. Add new vertices $v_{n(F)+1},\\ldots, v_p$. For each pair of vertices $v_i,v_j$ not covered in $F$, add"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.03461","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":""},"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"}