{"paper":{"title":"On the exact maximum induced density of almost all graphs and their inducibility","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Raphael Yuster","submitted_at":"2018-01-03T15:22:10Z","abstract_excerpt":"Let $H$ be a graph on $h$ vertices. The number of induced copies of $H$ in a graph $G$ is denoted by $i_H(G)$. Let $i_H(n)$ denote the maximum of $i_H(G)$ taken over all graphs $G$ with $n$ vertices.\n  Let $f(n,h) = \\Pi_{i}^h a_i$ where $\\sum_{i=1}^h a_i = n$ and the $a_i$ are as equal as possible. Let $g(n,h) = f(n,h) + \\sum_{i=1}^h g(a_i,h)$. It is proved that for almost all graphs $H$ on $h$ vertices it holds that $i_H(n)=g(n,h)$ for all $n \\le 2^{\\sqrt{h}}$. More precisely, we define an explicit graph property ${\\cal P}_h$ which, when satisfied by $H$, guarantees that $i_H(n)=g(n,h)$ for a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.01047","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"}