On the weight of Berge-F-free hypergraphs

Abstract

For a graph F, we say a hypergraph is a Berge-F if it can be obtained from F by replacing each edge of F with a hyperedge containing it. A hypergraph is Berge-F-free if it does not contain a subhypergraph that is a Berge-F. The weight of a non-uniform hypergraph H is the quantity P h2E(H) jhj. Suppose H is a Berge-F-free hypergraph on n vertices. In this short note, we prove that as long as every edge of H has size at least the Ramsey number of F and at most o(n), the weight of H is o(n2). This result is best possible in some sense. Along the way, we study other weight functions, and strengthen results of Gerbner and Palmer; and Grosz, Methuku and Tompkins.