On the cover Tur\'an number of Berge hypergraphs

For a fixed set of positive integers $R$, we say $\mathcal{H}$ is an $R$-uniform hypergraph, or $R$-graph, if the cardinality of each edge belongs to $R$. For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a Berge-$G$, denoted by $BG$, if there exists a bijection $f: E(G) \to E($\mathcal{H}$)$suchthatforevery$e \in E(G)$,$e \subseteq f(e)$. In this paper, we define a variant of Tur'an number in hypergraphs, namely the cover Tur'an number, denoted as ${\widehat{ex}}_R(n,G)$, as the maximum number of edges in the shadow graph of a Berge-$G$ free $R$-graph on $n$ vertices. We show a general upper bound on the cover Tur'an number of graphs and determine the cover Tur'an density of all graphs when the uniformity of the host hypergraph equals to $3$.