  For positive integers $s$, $t$, $m$ and $n$, the Zarankiewicz number
$Z_{s,t}(m,n)$ is defined to be the maximum number of edges in a bipartite
graph with parts of sizes $m$ and $n$ that has no complete biparitite subgraph
containing $s$ vertices in the part of size $m$ and $t$ vertices in the part of
size $n$. A simple argument shows that, for each $t \geq 2$,
$Z_{2,t}(m,n)=(t-1)\binom{m}{2}+n$ when $n \geq (t-1)\binom{m}{2}$. Here, for
large $m$, we determine the exact value of $Z_{2,t}(m,n)$ in almost all of the
remaining cases where $n=\Theta(tm^2)$. We establish a new family of upper
bounds on $Z_{2,t}(m,n)$ which complement a family already obtained by Roman.
We then prove that the floor of the best of these bounds is almost always
achieved. We also show that there are cases in which this floor cannot be
achieved and others in which determining whether it is achieved is likely a
very hard problem. Our results are proved by viewing the problem through the
lens of linear hypergraphs and our constructions make use of existing results
on edge decompositions of dense graphs.
