A hypergraph bipartite Turán problem with odd uniformity

Abstract

In this paper, we investigate the hypergraph Turán number $ex(n,K_{s,t}^{(r)})$. Here, $K_{s,t}^{(r)}$ 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 $ex(n,K_{s,t}^{(r)})=O_{s,t}(n^{r-\frac{1}{s-1}})$ 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 {\it even} and $t\gg s\ge 2$. Furthermore, they proved that $ex(n,K_{s,t}^{(3)})=O_{s,t}(n^{3-\frac{1}{s-1}-{\epsilon }_s})$ 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{equation*} \mbox{$ex(n,K_{s,t}^{(r)})=O_{r,s,t}(n^{r-\frac{1}{s-1}-\epsilon })$ holds for odd $r\ge 5$ and any $s\ge 3$.} \end{equation*} 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.