Asymptotics of the Hypergraph Bipartite Turán Problem

For positive integers s, t, r, let Ks,t(r) denote the r-uniform hypergraph whose vertex set is the union of pairwise disjoint sets X, Y₁, ⋯ , Y_t , where | X| = s and | Y₁| = ⋯ = | Y_t| = r- 1 , and whose edge set is { { x} ∪ Y_i: x∈ X, 1 ≤ i≤ t} . The study of the Turán function of Ks,t(r) received considerable interest in recent years. Our main results are as follows. First, we show that ex(n,Ks,t(r))=Os,r(t1s-1nr-1s-1) for all s, t≥ 2 and r≥ 3 , improving the power of n in the previously best bound and resolving a question of Mubayi and Verstraëte about the dependence of ex(n,K2,t(3)) on t. Second, we show that (1) is tight when r is even and t≫ s . This disproves a conjecture of Xu, Zhang and Ge. Third, we show that (1) is not tight for r= 3 , namely that ex(n,Ks,t(3))=Os,t(n3-1s-1-εs) (for all s≥ 3). This indicates that the behaviour of ex(n,Ks,t(r)) might depend on the parity of r. Lastly, we prove a conjecture of Ergemlidze, Jiang and Methuku on the hypergraph analogue of the bipartite Turán problem for graphs with bounded degrees on one side. Our tools include a novel twist on the dependent random choice method as well as a variant of the celebrated norm graphs constructed by Kollár, Rónyai and Szabó.