The Ramsey-Tur\'{a}n problem for cliques

Abstract

An important question in extremal graph theory raised by Vera T. S\'os asks to determine for a given integer $t\ge 3$ and a given positive real number $\delta$ the asymptotically supremal edge density $f_t(\delta)$ that an $n$-vertex graph can have provided it contains neither a complete graph $K_t$ nor an independent set of size $\delta n$. Building upon recent work of Fox, Loh, and Zhao [The critical window for the classical Ramsey-Tur\'an problem, Combinatorica 35 (2015), 435-476], we prove that if $\delta$ is sufficiently small (in a sense depending on $t$), then \[ f_t(\delta)= \begin{cases} \frac{3t-10}{3t-4}+\delta-\delta^2 & \text{ if $t$ is even,} \cr \frac{t-3}{t-1}+\delta & \text{ if $t$ is odd.} \end{cases} \]