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Some Exact Ramsey-Turán Numbers

Let r be an integer, f(n) a function, and H a graph. Introduced by Erdős, Hajnal, Sós, and Szemerédi, the r-Ramsey-Turán number of H, RT_r(n, H, f(n)), is defined to be the maximum number of edges in an n-vertex, H-free graph G with α_r(G) <= f(n) where α_r(G) denotes the K_r-independence number of G. In this note, using isoperimetric properties of the high dimensional unit sphere, we construct graphs providing lower bounds for RT_r(n,K_{r+s},o(n)) for every 2 <= s <= r. These constructions are sharp for an infinite family of pairs of r and s. The only previous sharp construction was by Bollobás and Erd\Hos for r = s = 2.