A hypergraph bipartite Tur\'an problem with odd uniformity

In this paper, we investigate the hypergraph Tur\'an number ex (n, K^ (r) ₒ, ₓ). Here, K^ (r) ₒ, ₓ denotes the r-uniform hypergraph with vertex set (₈ ₓXᵢ) Y and edge set \Xᵢ \{y\: 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 Erdos and has since received substantial attention in research. Recent advancements by Bradac, Gishboliner, Janzer and Sudakov have greatly contributed to a better understanding of this problem. They proved that ex (n, Kₒ, ₓ^ (r) ) =Oₒ, ₓ (n^r-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, Kₒ, ₓ^ (3) ) =Oₒ, ₓ (n^3-1{s-1-ₛ}) holds for s 3 and some ₛ>0. Addressing this intriguing discrepancy between the behavior of this number for r=3 and the even cases, Bradac et al. post a question of whether equation* ex (n, Kₒ, ₓ^ (ₑ) ) = Oₑ, ₒ, ₓ (n^{r-1{s-1- }) 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.