Induced Turán problems and traces of hypergraphs

Abstract

Let F be a graph. We say that a hypergraph H contains an induced BergeF if the vertices of F can be embedded to H (e.g., V(F)⊆V(H)) and there exists an injective mapping f from the edges of F to the hyperedges of H such that f(xy)∩V(F)={x,y} holds for each edge xy of F. In other words, H contains F as a trace.Let exr(n,BindF) denote the maximum number of edges in an r-uniform hypergraph with no induced Berge F. Let ex(n,Kr,F) denote the maximum number of Kr’s in an F-free graph on n vertices. We show that these two Turán type functions are strongly related.