The generalized Turán number for K_3 in graphs without suspensions of a path on five vertices

Given graphs $H$ and $F$, the generalized Turán number $\text{ex}(n,H,F)$ is defined as the maximum number of copies of $H$ in an $n$-vertex graph that contains no copy of $F$. The suspension $\widehat{F}$ of a graph $F$ is obtained by adding a new vertex that is adjacent to every vertex of $F$. Mubayi and Mukherjee (2023, DM) conjectured that $\text{ex}(n,K_3,\widehat{P_k})=\lfloor \frac{k-2}{2}\rfloor \cdot \frac{n^2}{8}+o(n^2)$, where $P_k$ is a path on $k\ge 4$ vertices. Using the triangle removal lemma, they verified this conjecture for $k=4,5,6$. Later, Mukherjee (2024, DM) established the exact value $\text{ex}(n,K_3,\widehat{P_4})=\lfloor n^2\mathrm{/}8\rfloor$. In this paper, using the stability method, we determine the exact value of $\text{ex}(n,K_3,\widehat{P_5})$ by showing that for sufficiently large $n$, $\text{ex}(n,K_3,\widehat{P_5})=\lfloor n^2\mathrm{/}8\rfloor \mathrm{.}$