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Erd\\H{o}s, Hajnal, Simonovits, S\\'os, and Szemer\\'edi posed several open questions about RT_t(n,K_s,o(n)), among them finding the minimum s such that $RT_t(n,K_{t+s},o(n)) = \\Omega(n^2)$, where it is easy to see that $RT_t(n,K_{t+1},o(n)) = o(n^2)$. In this paper, we answer this question by proving that $RT_t(n,K_{t+2},o(n)) "},"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":"1109.4428","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-20T20:42:29Z","cross_cats_sorted":[],"title_canon_sha256":"f1aec992a8797f2ab91233089175250a40726f31af48c10ad5b80fcc94776cf3","abstract_canon_sha256":"57338ae21e39e57b1d398bb472dba3c25d66ef8f66e62e96d051b96e0a98fe95"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:17:34.426142Z","signature_b64":"GKEktZIh0+W5QTOQU5E4Dom9fQPSp+cGnAkpRolF6PYQXdn4oU+57MvRDs+lkja0LmVdNsncl/s4H2GBGiBKDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"043a85c15b361e265dbacf1a6788734687e4bc35113efcf3ab14382f9d1fa254","last_reissued_at":"2026-05-18T03:17:34.425561Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:17:34.425561Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Ramsey-Tur\\'an numbers of graphs and hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"John Lenz, J\\'ozsef Balogh","submitted_at":"2011-09-20T20:42:29Z","abstract_excerpt":"Let t be an integer, f(n) a function, and H a graph. Define the t-Ramsey-Tur\\'an number of H, RT_t(n, H, f(n)), to be the maximum number of edges in an n-vertex, H-free graph G where f(n) is larger than the maximum number of vertices in a $K_t$-free induced subgraph of G. Erd\\H{o}s, Hajnal, Simonovits, S\\'os, and Szemer\\'edi posed several open questions about RT_t(n,K_s,o(n)), among them finding the minimum s such that $RT_t(n,K_{t+s},o(n)) = \\Omega(n^2)$, where it is easy to see that $RT_t(n,K_{t+1},o(n)) = o(n^2)$. 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