On generalized Tur{á}n problems with bounded matching number and circumference

Let $\mathcal{F}$ be a family of graphs. The generalized Turán number $\mathrm{ex}(n,K_r,\mathcal{F})$ is the maximum number of $K_r$ in an $n$-vertex graph that does not contain any member of $\mathcal{F}$ as a subgraph. Recently, Alon and Frankl initiated the study of Turán problems with bounded matching number. In this paper, we determine the generalized Turán number of $C_{\ge k}$ with bounded matching number.