Abstract
Let $\mathcal{F}$ be an $r$-uniform hypergraph and $G$ be a multigraph. The
hypergraph $\mathcal{F}$ is a Berge-$G$ if there is a bijection $f: E(G)
\rightarrow E( \mathcal{F} )$ such that $e \subseteq f(e)$ for each $e \in
E(G)$. Given a family of multigraphs $\mathcal{G}$, a hypergraph $\mathcal{H}$
is said to be $\mathcal{G}$-free if for each $G \in \mathcal{G}$, $\mathcal{H}$
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 $K_{2,t}$-free. We also determine an asymptotic formula for the maximum
number of edges in a linear 3-uniform 3-partite hypergraph that is $\{C_3 ,
K_{2,3} \}$-free.