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In this paper we prove some sharp results on this generalization of Tur\\'an numbers, where our focus is for the graphs $T,H$ satisfying $\\chi(T)<\\chi(H)$. This can be dated back to Erd\\H{o}s, where he generalized the celebrated Tur\\'an's theorem by showing that for any $r\\geq m$, the Tur\\'an graph $T_r(n)$ uniquely attains $ex(n,K_m,K_{r+1})$. For general graphs $H$ with $\\chi(H)=r+1>m$, Alon and Shikhelman showed that $ex(n,K_m,H)=\\binom{r}{m}(\\frac{n}{r})^m+o(n^m)$. 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