A Hypergraph Bipartite Turán Problem with Odd Uniformity

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.