{"paper":{"title":"Sparse halves in dense triangle-free graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Liana Yepremyan, Sergey Norin","submitted_at":"2013-11-22T17:13:00Z","abstract_excerpt":"Erd\\H{o}s conjectured that every triangle-free graph $G$ on $n$ vertices contains a set of $\\lfloor n/2 \\rfloor$ vertices that spans at most $n^2 /50$ edges. Krivelevich proved the conjecture for graphs with minimum degree at least $\\frac{2}{5}n$. Keevash and Sudakov improved this result to graphs with average degree at least $\\frac{2}{5}n$. We strengthen these results by showing that the conjecture holds for graphs with minimum degree at least $\\frac{5}{14}n$ and for graphs with average degree at least $(\\frac{2}{5} - \\varepsilon)n$ for some absolute $\\varepsilon >0$. Moreover, we show that t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.5818","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":""},"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"}