Many triangles in C5-free graphs

In the present paper, we introduce a new approach and use it to prove that the maximum number of triangles in a $C_5$-free graph on n vertices is at most $\frac{1}{2\sqrt{6}}(1+o(1\left)\right)n^{3/2}$, improving an estimate of Ergemlidze and Methuku [4]. We also show that the maximum size of an induced-$C_4$-free and $C_5$-free graph on n vertices is at most $\frac{1}{\sqrt{6}}(1+o(1\left)\right)n^{3/2}$, also improving an estimate of Ergemlidze and Methuku [4].