{"paper":{"title":"Upper Tails for Cliques","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Bobby DeMarco, Jeff Kahn","submitted_at":"2011-11-29T04:25:14Z","abstract_excerpt":"With $\\xi_{k}=\\xi_{k}^{n,p}$ the number of copies of $K_k$ in the usual (Erd\\H{o}s-R\\'enyi) random graph $G(n,p)$, $p\\geq n^{-2/(k-1)}$ and $\\eta>0$, we show when $k>1$ $$\\Pr(\\xi_k> (1+\\eta)\\E \\xi_k) < \\exp [-\\gO_{\\eta,k} \\min\\{n^2p^{k-1}\\log(1/p), n^kp^{\\binom{k}{2}}\\}].$$ This is tight up to the value of the constant in the exponent."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.6687","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"}