Extremal Results for Berge Hypergraphs

Let G be a graph and H be a hypergraph both on the same vertex set. We say that a hypergraph H is a Berge-G if there is a bijection f : E(G) → E(H) such that for e ∈ E(G) we have e ⊂ 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 (i.e. the sum of the cardinality of each edge) 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 Kővári-Sós-Turán theorem.