The Ramsey-Turán problem for cliques

Abstract

An important question in extremal graph theory raised by Vera T. Sós asks to determine for a given integer $t\ge 3$ and a given positive real number $δ$ the asymptotically supremal edge density $f_t(δ)$ that an $n$-vertex graph can have provided it contains neither a complete graph $K_t$ nor an independent set of size $δn$. Building upon recent work of Fox, Loh, and Zhao [The critical window for the classical Ramsey-Turán problem, Combinatorica 35 (2015), 435-476], we prove that if $δ$ is sufficiently small (in a sense depending on $t$), then \[ f_t(δ)= \begin{cases} \frac{3t-10}{3t-4}+δ-δ^2 & \text{ if $t$ is even,} \cr \frac{t-3}{t-1}+δ& \text{ if $t$ is odd.} \end{cases} \]