Infinite Tur\'an problems for bipartite graphs

We consider an infinite version of the bipartite Tur\'{a}n problem. Let $G$ be an infinite graph with $V(G) = \mathbb{N}$ and let $G_n$ be the $n$-vertex subgraph of $G$ induced by the vertices $\{1,2, \dots, n \}$. We show that if $G$ is $K_{2,t+1}$-free then for infinitely many $n$, $e(G_n) \leq 0.471 \sqrt{t} n^{3/2}$. Using the $K_{2,t+1}$-free graphs constructed by F\"{u}redi, we construct an infinite $K_{2,t+1}$-free graph with $e(G_n) \geq 0.23 \sqrt{t}n^{3/2}$ for all $n \geq n_0$.