On the Ramsey-Turán number with small $s$-independence number

Let $s$ be an integer, $f=f(n)$ a function, and $H$ a graph. Define the Ramsey-Turán number $RT_s(n,H, f)$ as the maximum number of edges in an $H$-free graph $G$ of order $n$ with $α_s(G) < f$, where $α_s(G)$ is the maximum number of vertices in a $K_s$-free induced subgraph of $G$. The Ramsey-Turán 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 $RT_s(n,K_t, n^δ)$ for fixed $δ<1$. We show that for an arbitrarily small $\varepsilon>0$ and $1/2<δ< 1$, $RT_s(n,K_{s+1}, n^δ) = Ω(n^{1+δ-\varepsilon})$ for all sufficiently large $s$. This is nearly optimal, since a trivial upper bound yields $RT_s(n,K_{s+1}, n^δ) = O(n^{1+δ})$. Furthermore, the range of $δ$ is as large as possible. We also consider more general cases and find bounds on $RT_s(n,K_{s+r},n^δ)$ for fixed $r\ge2$. Finally, we discuss a phase transition of $RT_s(n, K_{2s+1}, f)$ extending some recent result of Balogh, Hu and Simonovits.

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