Hypergraph Based Berge Hypergraphs

Fix a hypergraph F. A hypergraph H is called a Berge copy of F or Berge-F if we can choose a subset of each hyperedge of H to obtain a copy of F. A hypergraph H is Berge-F-free if it does not contain a subhypergraph which is Berge copy of F. This is a generalization of the usual, graph-based Berge hypergraphs, where F is a graph. In this paper, we study extremal properties of hypergraph based Berge hypergraphs and generalize several results from the graph-based setting. In particular, we show that for any r-uniform hypergraph F, the sum of the sizes of the hyperedges of a (not necessarily uniform) Berge-F-free hypergraph H on n vertices is o(n^r) when all the hyperedges of H are large enough. We also give a connection between hypergraph based Berge hypergraphs and generalized hypergraph Turán problems.