{"paper":{"title":"Two conjectures in Ramsey-Tur\\'an theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hong Liu, Jaehoon Kim, Younjin Kim","submitted_at":"2018-03-13T10:45:20Z","abstract_excerpt":"Given graphs $H_1,\\ldots, H_k$, a graph $G$ is $(H_1,\\ldots, H_k)$-free if there is a $k$-edge-colouring $\\phi:E(G)\\rightarrow [k]$ with no monochromatic copy of $H_i$ with edges of colour $i$ for each $i\\in[k]$. Fix a function $f(n)$, the Ramsey-Tur\\'an function $\\textrm{RT}(n,H_1,\\ldots,H_k,f(n))$ is the maximum number of edges in an $n$-vertex $(H_1,\\ldots,H_k)$-free graph with independence number at most $f(n)$. We determine $\\textrm{RT}(n,K_3,K_s,\\delta n)$ for $s\\in\\{3,4,5\\}$ and sufficiently small $\\delta$, confirming a conjecture of Erd\\H{o}s and S\\'os from 1979. It is known that $\\tex"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.04721","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"}