Some Exact Ramsey-Tur\'an Numbers

TL;DR

This paper establishes new lower bounds for the r-Ramsey-Turán numbers of complete graphs using geometric constructions, achieving sharp results for certain parameter pairs.

Contribution

It introduces a novel geometric approach to construct graphs that provide lower bounds for RT_r(n, K_{r+s}, o(n)), extending previous sharp results.

Findings

Constructed graphs using isoperimetric properties of high-dimensional spheres.

Provided lower bounds for RT_r(n, K_{r+s}, o(n)) for 2 <= s <= r.

Achieved sharpness of bounds for an infinite family of r and s.

Abstract

Let r be an integer, f(n) a function, and H a graph. Introduced by Erd\H{o}s, Hajnal, S\'{o}s, and Szemer\'edi, the r-Ramsey-Tur\'{a}n 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 \alpha_r(G) <= f(n) where \alpha_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 Bollob\'as and Erd\Hos for r = s = 2.