{"paper":{"title":"On the Profile of Multiplicities of Complete Subgraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Anne Kenyon, Shimon Kogan, Uriel Feige","submitted_at":"2017-03-28T17:32:56Z","abstract_excerpt":"Let $G$ be a $2$-coloring of a complete graph on $n$ vertices, for sufficiently large $n$. We prove that $G$ contains at least $n^{(\\frac{1}{4} - o(1))\\log n}$ monochromatic complete subgraphs of size $r$, where \\[ 0.3\\log n < r < 0.7\\log n. \\] The previously known lower bound on the total number of monochromatic complete subgraphs, due to Sz\\'{e}kely was $n^{0.1576\\log n}$. We also prove that $G$ contains at least $n^{\\frac{1}{7} \\log n} $ monochromatic complete subgraphs of size $\\frac{1}{2}\\log n$.\n  If furthermore one assumes that the largest monochromatic complete subgraph in $G$ is of si"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.09682","kind":"arxiv","version":3},"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"}