Turán numbers for Berge-hypergraphs and related extremal problems

Let F be a graph. We say that a hypergraph H is a Berge-F if there is a bijection f:E(F)→E(H) such that e⊆f(e) for every e∈E(F). Note that Berge-F actually denotes a class of hypergraphs. The maximum number of edges in an n-vertex r-graph with no subhypergraph isomorphic to any Berge-F is denoted ex _r (n,Berge-F). In this paper, we investigate the case when F=K _s,t and establish an upper-bound when r≥3, and a lower-bound when r=4 and t is large enough compared to s. Additionally, we prove a counting result for r-graphs of girth five that complements the asymptotic formula ex ₃ (n,Berge-{C ₂ ,C ₃ ,C ₄ })=[Formula presented]n ^3∕2 +o(n ^3∕2 ) of Lazebnik and Verstraëte (2003).