Generalized Turán problem for a path and a clique

Abstract

Let $H$ be a family of graphs. The generalized Turán number $\mathrm{ex}(n,K_r,H)$ is the maximum number of copies of the clique $K_r$ in any $n$-vertex $H$-free graph. In this paper, we determine the value of $\mathrm{ex}(n,K_r,\{P_k,K_m\})$ for sufficiently large $n$ with an exceptional case, and characterize all corresponding extremal graphs. This generalizes and strengthens the results of Katona (2024) on $\mathrm{ex}(n,K_2,\{P_k,K_m\})$. For the exceptional case, we obtain a tight upper bound for $\mathrm{ex}(n,K_r,\{P_k,K_m\})$ that confirms a conjecture on $\mathrm{ex}(n,K_2,\{P_k,K_m\})$ posed by Katona and Xiao.