The ACO Seminar (2023–2024)

In this paper, we investigate the hypergraph Turán number $\operatorname{ex}(n,K^{(r)}_{s,t})$. 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 Brada\v{c}, Gishboliner, Janzer and Sudakov have greatly contributed to a better understanding of this problem. They proved that $\operatorname{ex}(n,K_{s,t}^{(r)})=O_{s,t}(n^{r-\frac{1}{s-1}})$ holds for any $r\geq 3$ and $s,t\geq 2$. They also provided constructions illustrating the tightness of this bound if $r\geq 4$ is {\it even} and $t\gg s\geq 2$. Furthermore, they proved that $\operatorname{ex}(n,K_{s,t}^{(3)})=O_{s,t}(n^{3-\frac{1}{s-1}-\varepsilon_s})$ holds for $s\geq 3$ 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   $\operatorname{ex}(n,K_{s,t}^{(r)})= O_{r,s,t}(n^{r-\frac{1}{s-1}- \varepsilon})$ holds for odd $r\geq 5$ and any $s\geq 3$. 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.