Some Exact Ramsey-Turán Numbers
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چکیده
Let r be an integer, f(n) a function, and H a graph. Introduced by Erdős, Hajnal, Sós, and Szemerédi [8], the r-Ramsey-Turán 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 Bollobás and Erdős [6] for r = s = 2.
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تاریخ انتشار 2011