Some remarks on the Zarankiewicz problem

TL;DR

This paper refines bounds on the Zarankiewicz problem, demonstrating that a classical upper bound is nearly tight for many parameters using a novel algebraic probabilistic approach.

Contribution

It introduces a new quantitative variant of the random algebraic method to establish tight bounds for the Zarankiewicz problem.

Findings

Classical upper bound is tight up to a constant for many parameters.

New probabilistic algebraic technique developed for combinatorial bounds.

Improves understanding of extremal matrix configurations avoiding large all-ones submatrices.

Abstract

The Zarankiewicz problem asks for an estimate on $z(m, n; s, t)$, the largest number of $1$'s in an $m \times n$ matrix with all entries $0$ or $1$ containing no $s \times t$ submatrix consisting entirely of $1$'s. We show that a classical upper bound for $z(m, n; s, t)$ due to K\H{o}v\'ari, S\'os and Tur\'an is tight up to the constant for a broad range of parameters. The proof relies on a new quantitative variant of the random algebraic method.