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

Let $\cal H$ be a family of graphs. The Tur\'an number ${\rm ex}(n,{\cal H})$ is the maximum possible number of edges in an $n$-vertex graph which does not contain any member of $\cal H$ as a subgraph. As a common generalization of Tur\'an's theorem and Erd\H{o}s-Gallai theorem on the Tur\'an number of matchings, Alon and Frankl determined ${\rm ex}(n,{\cal H})$ for ${\cal H}=\{K_r,M_k\}$, where $M_k$ is a matching of size $k$. Replacing $M_k$ by $P_k$, Katona and Xiao obtained the Tur\'an number of ${\cal H}=\{K_r,P_k\}$ for $r \leq \lfloor k/2 \rfloor$ and sufficiently large $n$. In addition, they proposed a conjecture for the case of $r \geq \lfloor k/2 \rfloor+1$ and sufficiently large $n$. Motivated by the fact that the result for ${\rm ex}(n,P_k)$ can be deduced from the one for ${\rm ex}(n,{\cal C}_{\geq k})$, we investigate the Tur\'an number of ${\cal H}=\{K_r, {\cal C}_{\geq k}\}$ in this paper. 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 ${\cal H}=\{K_r, {\cal C}_{\geq k}\}$, we are able to show the value of ${\rm ex}(n,{\cal H})$ for $r \geq \lfloor (k-1)/2\rfloor+2$ and all $n$. As an application of this result, we confirm Katona and Xiao's conjecture in a stronger form. For $r \leq \lfloor (k-1)/2\rfloor+1$, we manage to show the value of ${\rm ex}(n,{\cal H})$ for sufficiently large $n$.