The Turán number of Berge-linear forests in hypergraphs

Let F be a family of graphs, and let H be a hypergraph. H is called a Berge-F if for some F∈F, there exists an injection θ:V(F)→V(H) and a bijection ϕ:E(F)→E(H) such that {θ(u),θ(v)}⊆ϕ(e) for each e={u,v}∈E(F). H is called Berge-F-free if H contains no subhypergraph isomorphic to any Berge-F. The Turán number of a Berge-F, denoted by ex_r(n,Berge-F), is defined as the maximum number of edges in an n -vertex Berge-F-free r -uniform hypergraph. A linear forest is a graph all components of which are paths or isolated vertices. Denote by L_n,k the family of all linear forests containing n vertices and k edges. In this paper, we determine the value of ex_r(n,Berge-L_n,k) for the cases 3≤r≤⌈k+12⌉−2 and r≥k+1. Furthermore, we characterize the extremal hypergraphs for the cases 3≤r≤k+12−3 and r≥k+1, when k is odd, and for the cases 3≤r≤k2−2 and r≥k+1, when k is even. We establish an upper bound on ex_r(n,Berge-L_n,k) for several other cases. Our results extend recently published results about the Turán numbers of Berge-matchings and linear forests.