Year: 2012
Type: article
Abstract: Let r be an integer, f(n) be a function, and H be a graph. Introduced by Erdős, Hajnal, Sós, and Szemerédi, 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 (for such values of r and s) was by Bollobás and Erdős for r=s=2.
DOI: 10.1112/blms/bds053
Authors: József Balogh, John Lenz
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2111794351