{"paper":{"title":"Vertex-Based Localization of Tur\\'{a}n's Theorem","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Bangalore), L. Sunil Chandran (Indian Institute of Science, Rajat Adak","submitted_at":"2025-04-03T17:51:50Z","abstract_excerpt":"Let $G$ be a simple graph with $n$ vertices and $m$ edges. According to Tur\\'{a}n's theorem, if $G$ is $K_{r+1}$-free, then $m \\leq |E(T(n, r))|,$ where $T(n, r)$ denotes the Tur\\'{a}n graph on $n$ vertices with a maximum clique of order $r$. A limitation of this statement is that it does not give an expression in terms of $n$ and $r$. A widely used version of Tur\\'{a}n's theorem states that for an $n$-vertex $K_{r+1}$-free graph, $m \\leq \\left\\lfloor \\frac{n^2(r-1)}{2r} \\right\\rfloor.$ Though this bound is often more convenient, it is not the same as the original statement. In particular, the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.02806","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2504.02806/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"}