EXTREMAL RESULTS FOR BERGE HYPERGRAPHS

Let E(G) and V(G) denote the edge set and vertex set of a (hyper) graph G. Let G be a graph and H be a hypergraph. We say that a hypergraph H is a Berge-G if there is a bijection f : E(G) -> E(H) such that for each e is an element of E(G) we have e subset of f(e). This generalizes the established definitions of "Berge path" and "Berge cycle" to general graphs. For a fixed graph G we examine the maximum possible size of a hypergraph with no Berge-G as a subhypergraph. In the present paper we prove general bounds for this maximum when G is an arbitrary graph. We also consider the specific case when G is a complete bipartite graph and prove an analogue of the Kovari-Sos-Turan theorem. In case G is C-4, we improve the bounds given by Gyori and Lemons