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The suspension of the path $P_4$ consists of a $P_4$ and an additional vertexconnected to each of the four vertices, and is denoted by $\hat{P_4}$. Thelargest number of triangles in a $\hat{P_4}$-free $n$-vertex graph is denotedby $ex(n,K_3,\hat{P_4})$. Mubayi and Mukherjee in 2020 showed that $ex(n,K_3,\hat{P_4})= n^2/8+O(n)$. We show that for sufficiently large $n$,$ex(n,K_3,\hat{P_4})=\lfloor n^2/8\rfloor$.

A note on the number of triangles in graphs without the suspension of a path on four vertices