The Zarankiewicz number $z(m,n;s,t)$ for a forbidden $K_{s,t}$ in a bipartite graph

For $m,n\in \mathbb{N}$ and $s,t\ge 1$, the Zarankiewicz number $z(m,n;s,t)$ is the maximum number of edges in a bipartite graph with a specified left part of size $m$ and right part of size $n$ that contains no ordinary $K_{s,t}$ whose $s$-vertex part lies on the left and whose $t$-vertex part lies on the right.

The edgeless bipartite graph makes the maximizing family nonempty, and only finitely many subsets of the $mn$ possible cross edges occur. Interchanging the two sides gives the exact symmetry

$z(m,n;s,t)=z(n,m;t,s)\mathrm{.}$

The orientation of $s,t$ is part of the notation; it will determine which additive term appears in the Kővári–Sós–Turán bound.