Hypergraph based Berge hypergraphs

Fix a hypergraph $\mathcal{F}$. A hypergraph $\mathcal{H}$ is called a {\it Berge copy of $\mathcal{F}$} or {\it Berge-$\mathcal{F}$} if we can choose a subset of each hyperedge of $\mathcal{H}$ to obtain a copy of $\mathcal{F}$. A hypergraph $\mathcal{H}$ is {\it Berge-$\mathcal{F}$-free} if it does not contain a subhypergraph which is Berge copy of $\mathcal{F}$. This is a generalization of the usual, graph based Berge hypergraphs, where $\mathcal{F}$ 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 hypregraph $\mathcal{F}$, the sum of the sizes of the hyperedges of a (not necessarily uniform) Berge-$\mathcal{F}$-free hypergraph $\mathcal{H}$ on $n$ vertices is $o(n^r)$ when all the hyperedges of $\mathcal{H}$ are large enough. We also give a connection between hypergraph based Berge hypergraphs and generalized hypergraph Tur\'an problems.