{"paper":{"title":"Positive codegree Andr\\'{a}sfai--Erd\\H{o}s--S\\'{o}s theorem for the generalized triangle","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jian Wang, Sijie Ren, Xizhi Liu","submitted_at":"2024-11-11T16:12:09Z","abstract_excerpt":"The celebrated Andr\\'{a}sfai--Erd\\H{o}s--S\\'{o}s Theorem from 1974 shows that every $n$-vertex triangle-free graph with minimum degree greater than $2n/5$ must be bipartite.\n  We establish a positive codegree extension of this result for the $r$-uniform generalized triangle $\\mathrm{T}_{r} = \\left\\{\\{1,\\ldots, r-1,r\\}, \\{1,\\ldots, r-1,r+1\\},\\{r,r+1, \\ldots, 2r-1\\}\\right\\}$$\\colon$ For every $n \\ge (r-1)(2r+1)/2$, if $\\mathcal{H}$ is an $n$-vertex $\\mathrm{T}_{r}$-free $r$-uniform hypergraph in which each $(r-1)$-tuple of vertices is contained in either zero edges or more than $2n/(2r+1)$ edges"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.07090","kind":"arxiv","version":2},"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/2411.07090/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"}