{"paper":{"title":"Edge-statistics on large graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Dan Hefetz, Michael Krivelevich, Mykhaylo Tyomkyn, Noga Alon","submitted_at":"2018-05-17T16:35:01Z","abstract_excerpt":"The inducibility of a graph $H$ measures the maximum number of induced copies of $H$ a large graph $G$ can have. Generalizing this notion, we study how many induced subgraphs of fixed order $k$ and size $\\ell$ a large graph $G$ on $n$ vertices can have. Clearly, this number is $\\binom{n}{k}$ for every $n$, $k$ and $\\ell \\in \\left \\{0, \\binom{k}{2} \\right\\}$. We conjecture that for every $n$, $k$ and $0 < \\ell < \\binom{k}{2}$ this number is at most $\\left(1/e + o_k(1) \\right) \\binom{n}{k}$. If true, this would be tight for $\\ell \\in \\{1, k-1\\}$.\n  In support of our `Edge-statistics conjecture' "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.06848","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1805.06848/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}