In this paper, we investigate the hypergraph Turán number \(\textrm{ex}(n,K^{(r)}_{s,t})\) . Here, \(K^{(r)}_{s,t}\) denotes the r-uniform hypergraph with vertex set \(\left( \cup _{i\in [t]}X_i\right) \cup Y\) and edge set \(\{X_i\cup \{y\}: i\in [t], y\in Y\}\) , where \(X_1,X_2,\cdots ,X_t\) are t pairwise disjoint sets of size \(r-1\) and Y is a set of size s disjoint from each \(X_i\) . This study was initially explored by Erdős and has since received substantial attention in research. Recent advancements by Bradač, Gishboliner, Janzer and Sudakov have greatly contributed to a better understanding of this problem. They proved that \(\textrm{ex}(n,K_{s,t}^{(r)})=O_{s,t}(n^{r-\frac{1}{s-1}})\) holds for any \(r\ge 3\) and \(s,t\ge 2\) . They also provided constructions illustrating the tightness of this bound if \(r\ge 4\) is even and \(t\gg s\ge 2\) . Furthermore, they proved that \(\textrm{ex}(n,K_{s,t}^{(3)})=O_{s,t}(n^{3-\frac{1}{s-1}-\varepsilon _s})\) holds for \(s\ge 3\) and some \(\epsilon _s>0\) . Addressing this intriguing discrepancy between the behavior of this number for \(r=3\) and the even cases, Bradač et al. post a question of whether \(\begin{aligned} \textrm{ex}(n,K_{s,t}^{(r)})= O_{r,s,t}(n^{r-\frac{1}{s-1}- \varepsilon }) \text{ holds } \text{ for } \text{ odd } r\ge 5 \text{ and } \text{ any } s\ge 3\text{. } \end{aligned}\) In this paper, we provide an affirmative answer to this question, utilizing novel techniques to identify regular and dense substructures. This result highlights a rare instance in hypergraph Turán problems where the solution depends on the parity of the uniformity.
