Improved bounds for the extremal number of subdivisions

Let $H_t$ be the subdivision of $K_t$. Very recently, Conlon and Lee have proved that for any integer $t\ge 3$, there exists a constant $C$ such that $\text{ex}(n,H_t)\le C{n}^{3\mathrm{/}2-1\mathrm{/}{6}^t}$. In this paper, we prove that there exists a constant $C^{\mathrm{\prime }}$ such that $$\text{ex}$(n,H_t)\leq C’n^{3/2-\frac{1}{4t-6}}$.