On the Turán number of 1-subdivision of $K_{3,t}$

Abstract

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

15 pages