Exact values for some unbalanced Zarankiewicz numbers

Abstract

For positive integers $s$, $t$, $m$ and $n$, the Zarankiewicz number $Z_{s,t}\left(m,n\right)$ is defined to be the maximum number of edges in a bipartite graph with parts of sizes $m$ and $n$ that has no complete bipartite 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 $t\ge 2$, $Z_{2,t}\left(m,n\right)=\left(t-1\right)\left(\frac{m}{2}\right)+n$ when $n\ge \left(t-1\right)\left(\frac{m}{2}\right)$. Here, for large $m$, we determine the exact value of $Z_{2,t}\left(m,n\right)$ in almost all of the remaining cases where $n=\Theta \left(t{m}^2\right)$. We establish a new family of upper bounds on $Z_{2,t}\left(m,n\right)$ 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.