A hypergraph bipartite Tur\'an problem with odd uniformity

In this paper, we investigate the hypergraph Tur\'an number ex(n,K⁽ʳ⁾s,t). Here, K⁽ʳ⁾s,t denotes the r-uniform hypergraph with vertex set (∪i∈ [t]Xᵢ)∪ Y and edge set ᵢ∪ \: i∈ [t], y∈ Y\, where X₁,X₂,⋯,Xₜ are t pairwise disjoint sets of size $r-1$ and Y is a set of size s disjoint from each Xᵢ. This study was initially explored by Erd{o}s and has since received substantial attention in research. Recent advancements by Brada{c}, Gishboliner, Janzer and Sudakov have greatly contributed to a better understanding of this problem. They proved that ex(n,Ks,t⁽ʳ⁾)=Os,t(nr-1/s-1) holds for any r≥ 3 and s,t≥ 2. They also provided constructions illustrating the tightness of this bound if r≥ 4 is { even} and t s≥ 2. Furthermore, they proved that ex(n,Ks,t⁽³⁾)=Os,t(n3-1/s-1-εₛ) holds for s≥ 3 and some εₛ>0. Addressing this intriguing discrepancy between the behavior of this number for $r=3$ and the even cases, Brada{c} et al. post a question of whether {equation*} ex(n,Ks,t⁽ʳ⁾)= Or,s,t(nr-1/s-1- ε) holds for odd r≥ 5 and any s≥ 3. {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 Tur\'an problems where the solution depends on the parity of the uniformity.