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

Let $s$ be an integer, $f=f(n)$ a function, and $H$ a graph. Define the Ramsey-Tur'an number $R{T}_s(n,H,f)$ as the maximum number of edges in an $H$-free graph $G$ of order $n$ with ${\alpha }_s(G)<f$, where ${\alpha }_s(G)$ is the maximum number of vertices in a $K_s$-free induced subgraph of $G$. The Ramsey-Tur'an number attracted a considerable amount of attention and has been mainly studied for $f$ not too much smaller than $n$. In this paper we consider $R{T}_s(n,K_t,n^{\delta })$ for fixed $\delta <1$. We show that for an arbitrarily small $\epsilon >0$ and $1\mathrm{/}2<\delta <1$, $RT_s(n,K_{s+1}, n^{\delta}) = \Omega(n^{1+\delta-\varepsilon})$forallsufficientlylarge$s$. This is nearly optimal, since a trivial upper bound yields $RT_s(n,K_{s+1}, n^{\delta}) = O(n^{1+\delta})$\mathrm{.}Furthermore,therangeof$\delta$ is as large as possible. We also consider more general cases and find bounds on $R{T}_s(n,K_{s+r},n^{\delta })$ for fixed $r\ge 2$. Finally, we discuss a phase transition of $R{T}_s(n,K_{2s+1},f)$ extending some recent result of Balogh, Hu and Simonovits.