On $r$-uniform linear hypergraphs with no Berge-$K_{2,t}$

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( $\mathcal{F}$ )$suchthat$e \subseteq f(e)$foreach$e \in E(G)$\mathrm{.}Givenafamilyofmultigraphs$$\mathcal{G}$$,ahypergraph$$\mathcal{H}$$issaidtobe$$\mathcal{G}$$-freeifforeach$G \in $\mathcal{G}$$,$$\mathcal{H}$$doesnotcontainasubhypergraphthatisisomorphictoaBerge-$G$. 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.