Some exact Ramsey-Turán numbers

József Balogh,John Lenz

doi:10.1112/blms/bds053

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.

Highlights

Let G be a graph and define the Kr-independence number of G as αr(G) := max {|S| : S ⊆ V (G), G[S] is Kr-free} .Define RTr(n, H, f (n)) to be the maximum number of edges in an H-free graph G on n vertices with αr(G) ≤ f (n) and let θr (H ) = lim ǫ→0 lim n→∞ 1 n2 RTr(n, H, ǫn). (1)We write RTr(n, H, o(n)) = θr(H)n2 + o(n2)
Turan’s Theorem [18] states that the maximum number of edges in a Kr-free graph on n vertices is achieved by the complete (r − 1)-partite graph
This extremal graph has independent sets with linear size, which motivated Erdos and Sos [10] to ask about the maximum number of edges in a

Summary

Introduction

Let G be a graph and define the Kr-independence number of G as αr(G) := max {|S| : S ⊆ V (G), G[S] is Kr-free}. Define RTr(n, H, f (n)) to be the maximum number of edges in an H-free graph G on n vertices with αr(G) ≤ f (n) and let θr (H ). Sos, Hajnal, and Szemeredi [8] extended these results to determine θ2(K2r) for all r ≥ 2 Another Ramsey-Turan result is an important and widely applicable theorem of Ajtai, Komlos, and Szemeredi [2]. They lower bounded the independence number of triangle-free, n-vertex graphs with m edges. Their result can be phrased as RT2 n, K3, cn m log m n. The appendix contains a sketch of the proof of Proposition 3

Construction

Verifying properties of G

The Kr-independence number of G

Concluding Remarks

A Upper bounds