A hypergraph bipartite Turán problem with odd uniformity

In this paper, we investigate the hypergraph Turán number ex(n,K^(r)_s,t). Here, K^(r)_s,t denotes the r-uniform hypergraph with vertex set (∪_i∈ [t]X_i)∪ Y and edge set {X_i∪{y}: i∈ [t], y∈ Y}, where X_1,X_2,⋯,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-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,K_s,t^(3))=O_s,t(n^3-1/s-1-ε_s) holds for s≥ 3 and some ϵ_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 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.