Asymptotics of the Hypergraph Bipartite Turán Problem

For positive integers s , t , r , let K_s,t^(r) denote the r -uniform hypergraph whose vertex set is the union of pairwise disjoint sets X,Y_1,… ,Y_t , where |X| = s and |Y_1| = … = |Y_t| = r-1 , and whose edge set is {{x}∪ Y_i: x ∈ X, 1≤ i≤ t} . The study of the Turán function of K_s,t^(r) received considerable interest in recent years. Our main results are as follows. First, we show that 1 ex( n,K_s,t^(r)) = O_s,r( t^1/s-1n^r - 1/s-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,K_2,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,K_s,t^(3)) = O_s,t(n^3 - 1/s-1 - ε _s) (for all s≥ 3 ). This indicates that the behaviour of ex(n,K_s,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.