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For a graph $H$, the 1-subdivision of $H$, denoted by $H'$, is the graphobtained by replacing the edges of $H$ by internally disjoint paths of length2. Recently, Conlon, Janzer and Lee (arXiv: 1903.10631) asked the followingquestion: For any integer $s\ge2$, estimate the smallest $t$ such that$\textup{ex}(n,K_{s,t}')=\Omega(n^{\frac{3}{2}-\frac{1}{2s}})$. In this paper,we consider the case $s=3$. More precisely, we provide an explicit constructiongiving \begin{align*} \text{ex}(n,K_{3,30}')=\Omega(n^{\frac{4}{3}}),\end{align*} which reduces the estimation for the smallest value of $t$ from amagnitude of $10^{56}$ to the number $30$. The construction is algebraic, whichis based on some equations over finite fields.

On the Tur\'{a}n number of 1-subdivision of $K_{3,t}$