Maximum cliques in a graph without disjoint given subgraph

Abstract: The generalized Tur\'an number $\ex(n,K_s,F)$ denotes the maximum number of copies of $K_s$ in an $n$-vertex $F$-free graph. Let $kF$ denote $k$ disjoint copies of $F$. Gerbner, Methuku and Vizer [DM, 2019, 3130-3141] gave a lower bound for $\ex(n,K_3,2C_5)$ and obtained the magnitude of $\ex(n, K_s, kK_r)$. In this paper, we determine the exact value of $\ex(n,K_3,2C_5)$ and described the unique extremal graph for large $n$. Moreover, we also determine the exact value of $\ex(n,K_r,(k+1)K_r)$ which generalizes some known results.