On generalized Turán problems with bounded matching number

The generalized Turán number $\mathrm{e}\mathrm{x}(n,H,\mathcal{F})$ is defined as the maximum number of copies of a graph $H$ in an $n$-vertex graph that does not contain any graph $F\in \mathcal{F}$. Alon and Frankl initiated the study of Turán problems with a bounded matching number.In this paper, we establish stability results for generalized Turán problems with bounded matching number.Using the stability results, we provide exact values of $\text{ex}(n,K_r,\{F,M_{s+1}\})$ for $F$ being any non-bipartite graph or a path on $k$ vertices.