Some exact Ramsey–Turán numbers

Let r be an integer, f(n) a function, and H a graph. Introduced by Erdýos, Hajnal, Sos, and Szemeredi (8), the r-Ramsey-Turan number of H, RTr(n,H,f(n)), is defined to be the maximum number of edges in an n-vertex, H-free graph G with αr(G) ≤ f(n) where αr(G) denotes the Kr-independence number of G. In this note, using isoperimetric properties of the high dimensional unit sphere, we construct graphs providing lower bounds for RTr(n,Kr+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 Bollobas and Erdýos (6) for r = s = 2.