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Let $s$ be an integer, $f=f(n)$ a function, and $H$ a graph. Define theRamsey-Tur\'an number $RT_s(n,H, f)$ as the maximum number of edges in an$H$-free graph $G$ of order $n$ with $\alpha_s(G) < f$, where $\alpha_s(G)$ isthe maximum number of vertices in a $K_s$-free induced subgraph of $G$. TheRamsey-Tur\'an number attracted a considerable amount of attention and has beenmainly studied for $f$ not too much smaller than $n$. In this paper we consider$RT_s(n,K_t, n^{\delta})$ for fixed $\delta<1$. We show that for an arbitrarilysmall $\varepsilon>0$ and $1/2<\delta< 1$, $RT_s(n,K_{s+1}, n^{\delta}) =\Omega(n^{1+\delta-\varepsilon})$ for all sufficiently large $s$. This isnearly optimal, since a trivial upper bound yields $RT_s(n,K_{s+1}, n^{\delta})= O(n^{1+\delta})$. Furthermore, the range of $\delta$ is as large as possible.We also consider more general cases and find bounds on$RT_s(n,K_{s+r},n^{\delta})$ for fixed $r\ge2$. Finally, we discuss a phasetransition of $RT_s(n, K_{2s+1}, f)$ extending some recent result of Balogh, Huand Simonovits.

On the Ramsey-Tur\'an number with small $s$-independence number