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 + 1 2 ⌉ − 2 and r ≥ k + 1 . Furthermore, we characterize the extremal hypergraphs for the cases 3 ≤ r ≤ k + 1 2 − 3 and r ≥ k + 1 , when k is odd, and for the cases 3 ≤ r ≤ k 2 − 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.