On r-uniform linear hypergraphs with no Berge-K2,t

Let F be an r-uniform hypergraph and G be a multigraph. The hypergraph F is a Berge-G if there is a bijection f : E(G) -> E(F) such that e subset of f(e) for each e is an element of E(G). Given a family of multigraphs G, a hypergraph H is said to be G-free if for each G is an element of G, H does not contain a subhypergraph that is isomorphic to a Berge-G. We prove bounds on the maximum number of edges in an r-uniform linear hypergraph that is K-2,K-t-free. We also determine an asymptotic formula for the maximum number of edges in a linear 3-uniform 3-partite hypergraph that is {C-3, K-2,K-3}-free.