{"paper":{"title":"Some sharp results on the generalized Tur\\'an numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jie Ma, Yu Qiu","submitted_at":"2018-02-04T08:31:34Z","abstract_excerpt":"For graphs $T, H$, let $ex(n,T,H)$ denote the maximum number of copies of $T$ in an $n$-vertex $H$-free graph. In this paper we prove some sharp results on this generalization of Tur\\'an numbers, where our focus is for the graphs $T,H$ satisfying $\\chi(T)<\\chi(H)$. This can be dated back to Erd\\H{o}s, where he generalized the celebrated Tur\\'an's theorem by showing that for any $r\\geq m$, the Tur\\'an graph $T_r(n)$ uniquely attains $ex(n,K_m,K_{r+1})$. For general graphs $H$ with $\\chi(H)=r+1>m$, Alon and Shikhelman showed that $ex(n,K_m,H)=\\binom{r}{m}(\\frac{n}{r})^m+o(n^m)$. Here we determin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.01091","kind":"arxiv","version":1},"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"}