{"paper":{"title":"Dense Subgraphs in Random Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Alexander Veremyev, B\\'ela Bollob\\'as, Julian Sahasrabudhe, Paul Balister","submitted_at":"2018-03-27T22:39:17Z","abstract_excerpt":"For a constant $\\gamma \\in[0,1]$ and a graph $G$, let $\\omega_{\\gamma}(G)$ be the largest integer $k$ for which there exists a $k$-vertex subgraph of $G$ with at least $\\gamma\\binom{k}{2}$ edges. We show that if $0<p<\\gamma<1$ then $\\omega_{\\gamma}(G_{n,p})$ is concentrated on a set of two integers. More precisely, with $\\alpha(\\gamma,p)=\\gamma\\log\\frac{\\gamma}{p}+(1-\\gamma)\\log\\frac{1-\\gamma}{1-p}$, we show that $\\omega_{\\gamma}(G_{n,p})$ is one of the two integers closest to $\\frac{2}{\\alpha(\\gamma,p)}\\big(\\log n-\\log\\log n+\\log\\frac{e\\alpha(\\gamma,p)}{2}\\big)+\\frac{1}{2}$, with high probabi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.10349","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"}