Improved upper bounds on Zarankiewicz numbers

TL;DR

This paper improves upper bounds on Zarankiewicz numbers by enhancing the linear programming approach with additional constraints, leading to new bounds for small parameters and a generalized closed-form bound.

Contribution

It introduces new constraints to the linear program for Zarankiewicz numbers, resulting in tighter upper bounds and a generalized closed-form expression.

Findings

Improved upper bounds for many small parameter sets.

Established a new family of closed-form bounds.

Generalized previous results for the case s=2.

Abstract

For positive integers $s,t,m$ and $n$, the Zarankiewicz number $z(m,n;s,t)$ is the maximum number of edges in a subgraph of $K_{m,n}$ that has no complete bipartite subgraph containing $s$ vertices in the part of size $m$ and $t$ vertices in the part of size $n$. The best general upper bound on Zarankiewicz numbers is a bound due to Roman that can be viewed as the optimal value of a simple linear program. Here we show that in many cases this bound can be improved by adding additional constraints to this linear program. This allows us to prove new upper bounds on Zarankiewicz numbers for many small parameter sets. We are also able to establish a new family of closed form upper bounds on $z(m,n;s,t)$ that captures much, but not all, of the power of the new constraints. This bound generalises a recent result of Chen, Horsley and Mammoliti that applied only in the case $s=2$.