On the cover Turan number of Berge hypergraphs

For a fixed set of positive integers R, we say N 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 N is called a Berge -G, if there is an injection i: V(G) -> V(N) and a bijection f : E(G) -> E(N) such that for all e = uv E E(G), we have {i(u), i(v)} c f(e). We define the cover Turan number of a graph G, denoted as circumflex accent exR(n, BG), as the maximum number of edges in the 2-shadow of an n-vertex Berge -G-free R-graph, where the 2-shadow H of a hypergraph N is a graph such that an edge e E E(H) if and only if e c h for some h E E(N). In this paper, we show an Erdos-Stone-Simonovits-type upper bound on the cover Turan number of graphs and determine the cover Turan density of all graphs when the host hypergraph is 3-uniform. (c) 2021 Elsevier Ltd. All rights reserved.