The maximum number of triangles in $K_{1,s,t}$-free graphs

We consider the following generalized Turán problem: For $2 \le s \le t$, what is the maximum number of triangles in a $K_{1,s,t}$-free graph on $n$ vertices? The previously best known lower and upper bounds are $Ω(n^2)$ and $o(n^{3-1/s})$, respectively. To the best of our knowledge, all known proofs of the upper bound use the triangle removal lemma. We give a new elementary proof that avoids the use of the triangle removal lemma and improves the upper bound to $O\left(n^{3-1/s}(\log n)^{-1+1/s}\right)$.