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Some remarks on the Zarankiewicz problem
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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 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.
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Cited by 1 Pith paper
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A congruence obstruction to Roman's bound for Zarankiewicz numbers
A congruence argument shows Roman's bound is not tight on a long interval below the design threshold, with exact values in a special case.
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