On $r$-Uniform Linear Hypergraphs with no Berge-$K_{2, t}$

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 ⊆ f(e) for each e ∈ E(G). Given a family of multigraphs G, a hypergraph H is said to be G-free if for each G ∈ 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 K2,t-free. We also determine an asymptotic formula for the maximum number of edges in a linear 3-uniform 3-partite hypergraph that is {C3,K2,3}-free.