The number of edges in graphs with bounded clique number and circumference

Let $H$ be a family of graphs. The Turán number $\mathrm{ex}(n,H)$ is the maximum possible number of edges in an n-vertex graph which does not contain any member of $H$ as a subgraph. As a common generalization of Turán's theorem and Erdős-Gallai theorem on the Turán number of matchings, Alon and Frankl determined $\mathrm{ex}(n,H)$ for $H=\{K_r,M_k\}$, where $M_k$ is a matching of size k. Replacing $M_k$ by $P_k$, Katona and Xiao obtained the Turán number of $H=\{K_r,P_k\}$ for $r\le \lfloor k/2\rfloor$ and sufficiently large n. In addition, they proposed a conjecture for the case where $r\ge \lfloor k/2\rfloor +1$ and n is sufficiently large. Motivated by the fact that the result for $\mathrm{ex}(n,P_k)$ can be deduced from the one for $\mathrm{ex}(n,C_{\ge k})$, we investigate the Turán number of $\{K_r,C_{\ge k}\}$ in this paper, where $C_{\ge k}$ denotes the set of cycles of length at least k. In other words, we aim to determine the maximum number of edges in graphs with clique number at most $r-1$ and circumference at most $k-1$. For $H=\{K_r,C_{\ge k}\}$, we are able to show the value of $\mathrm{ex}(n,H)$ for $r\ge \lfloor (k-1)/2\rfloor +2$ and all n. As an application of this result, we confirm Katona and Xiao's conjecture in a stronger form. For $r\le \lfloor (k-1)/2\rfloor +1$, we manage to show the value of $\mathrm{ex}(n,H)$ for sufficiently large n.