  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 \[
f_t(\delta)=
  \begin{cases}
  \frac{3t-10}{3t-4}+\delta-\delta^2 & \text{ if $t$ is even,} \cr
  \frac{t-3}{t-1}+\delta & \text{ if $t$ is odd.}
  \end{cases} \]
