The Turán numbers of two kinds of Berge hypergraphs

Let F be a graph. A hypergraph mathcalHis called a Berge-F if there is a bijection phi: E(F)to E(mathcalH such that e⊆ phi(e) for each e∈ E(F). We say that the hypergraph mathcalHis Berge-F-free if there is no sub-hypergraph of mathcalHisomorphic to any Berge-F. The maximum number of edges, denoted by mboxexr(n, mboxBerge-F), in an n-vertex r-uniform Berge-F-free hypergraph is the Turán number of Berge-F. A graph F is called an edge-critical graph if the removal of an edge decreases the chromatic number of F. In this paper, we give the Turán number of Berge-F when F is pK_k or an edge-critical graph.