Asymptotics of the Hypergraph Bipartite Turán Problem

Abstract

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,dots ,Y_t, where |X| = s and |Y_1| = dots = |Y_t| = r-1, and whose edge set is {{x} cup Y_i: x in X, 1le ile 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 1exn,Ks,t(r)=Os,rt1s-1nr-1s-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \ extrm{ex}\\left( n,K_{s,t}^{(r)}\\right) = O_{s,r}\\left( t^{\\frac{1}{s-1}}n^{r - \\frac{1}{s-1}}\\right) \\end{aligned}$$\\end{document}for all s,tge 2 and rge 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)}) on t. Second, we show that (1) is tight when r is even and t gg s. This disproves a conjecture of Xu, Zhang and Ge. Third, we show that (1) is not tight for r = 3, namely that textrm{ex}(n,K_{s,t}^{(3)}) = O_{s,t}(n^{3 - frac{1}{s-1} - varepsilon _s}) (for all sge 3). This indicates that the behaviour of textrm{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.