Abstract. Given an [Formula: see text]-graph [Formula: see text] that satisfies certain properties, for all sufficiently large [Formula: see text] and for each [Formula: see text] in the interval [Formula: see text], where [Formula: see text] is a constant depending on [Formula: see text], we determine the maximum number of edges in an [Formula: see text]-vertex graph that does not contain [Formula: see text] vertex-disjoint copies of [Formula: see text]. In fact, our method can also apply to a rainbow version of the above result, including the characterization of extremal constructions. Moreover, it can be applied to determine the anti-Ramsey numbers. This generalizes a result of Hou et al. [ 19 , J. Combin. Theory Ser. B, 172 (2025), pp. 221–262], and further extends the work of Gan et al. [ 16 , J. Graph Theory, 104 (2023), pp. 516–556], Bushaw and Kettle [ 4 , SIAM J. Discrete Math., 28 (2014), pp. 711–721], Khormali and Palmer [ 24 , European J. Combin., 102 (2022), 103506], as well as several earlier results.