Exact values for unbalanced Zarankiewicz numbers

TL;DR

This paper determines the exact values of unbalanced Zarankiewicz numbers for most cases when n is proportional to t times m squared, introduces new upper bounds, and explores the complexity of achieving these bounds.

Contribution

It provides almost complete exact values for Z_{2,t}(m,n) in large m cases, introduces new upper bounds, and analyzes the attainability and complexity of these bounds.

Findings

Exact values of Z_{2,t}(m,n) determined for most cases with n=Θ(tm^2)

New upper bounds on Zarankiewicz numbers established

The floor of the bounds is almost always achieved, but not always, and some cases are computationally hard

Abstract

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.