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For given integers $2\\le s<t$ let $$ f_{s,t}(n)=\\min \\{\\max \\{|W| : W\\subseteq V(G) {and} G[W] {contains no} K_s\\} \\}, $$ where the minimum is taken over all $K_t$-free graphs $G$ of order $n$. In this paper, we show that for every $s\\ge 3$ there exist constants $c_1=c_1(s)$ and $c_2=c_2(s)$ such that $f_{s,s+1}(n) \\le c_1 (\\log n)^{c_2} \\sqrt{n}$. This result is best possible up to a polylogarithmic factor. 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