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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1803.04721","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-03-13T10:45:20Z","cross_cats_sorted":[],"title_canon_sha256":"3cf3bdb57644c2d97b568cd90a1c197a48f1f6f34c058e7c9fe0b47f75fda650","abstract_canon_sha256":"c30148098e789ea12877c6b032b8092d077a784ac619d316c46a6e22ffcebbdd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:21:18.032556Z","signature_b64":"AstDTzVqCTsGBmbWAER2B+lwH0rwqHJpWqLFqFG7GMiFbQeBsQqEMbXYDmVNf/K9z0VHP3Rv6j6nC5rqRWdOAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4000a1e06ecabc4f0df42b06273e63c0735da4ea1ab8137c428b9ba7bf7acd43","last_reissued_at":"2026-05-18T00:21:18.031865Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:21:18.031865Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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