Asymptotics of the Hypergraph Bipartite Turán Problem

Abstract

AbstractFor positive integers s, t, r, let $$K_{s,t}^{(r)}$$ K s , t ( r ) denote the r-uniform hypergraph whose vertex set is the union of pairwise disjoint sets $$X,Y_1,\dots ,Y_t$$ X , Y 1 , ⋯ , Y t , where $$|X| = s$$ | X | = s and $$|Y_1| = \dots = |Y_t| = r-1$$ | Y 1 | = ⋯ = | Y t | = r - 1 , and whose edge set is $$\{\{x\} \cup Y_i: x \in X, 1\le i\le t\}$$ { { x } ∪ Y i : x ∈ X , 1 ≤ i ≤ t } . The study of the Turán function of $$K_{s,t}^{(r)}$$ K s , t ( r ) received considerable interest in recent years. Our main results are as follows. First, we show that $$\begin{aligned} \textrm{ex}\left( n,K_{s,t}^{(r)}\right) = O_{s,r}\left( t^{\frac{1}{s-1}}n^{r - \frac{1}{s-1}}\right) \end{aligned}$$ ex n , K s , t ( r ) = O s , r t 1 s - 1 n r - 1 s - 1 for all $$s,t\ge 2$$ s , t ≥ 2 and $$r\ge 3$$ r ≥ 3 , improving the power of n in the previously best bound and resolving a question of Mubayi and Verstraëte about the dependence of $$\textrm{ex}(n,K_{2,t}^{(3)})$$ ex ( n , K 2 , t ( 3 ) ) on t. Second, we show that (1) is tight when r is even and $$t \gg s$$ t ≫ s . This disproves a conjecture of Xu, Zhang and Ge. Third, we show that (1) is not tight for $$r = 3$$ r = 3 , namely that $$\textrm{ex}(n,K_{s,t}^{(3)}) = O_{s,t}(n^{3 - \frac{1}{s-1} - \varepsilon _s})$$ ex ( n , K s , t ( 3 ) ) = O s , t ( n 3 - 1 s - 1 - ε s ) (for all $$s\ge 3$$ s ≥ 3 ). This indicates that the behaviour of $$\textrm{ex}(n,K_{s,t}^{(r)})$$ 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.