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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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