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arXiv · 2007.12816

Some remarks on the Zarankiewicz problem

David Conlon
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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 Kővári, Sós and Turán 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.

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David Conlon. 2021-04-11. Some remarks on the Zarankiewicz problem. https://doi.org/10.1017/s0305004121000475

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