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Some remarks on the Zarankiewicz problem

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  15 June 2021

DAVID CONLON
DAVID CONLON*
Affiliation:
Department of Mathematics, California Institute of Technology, 1200 E. California Boulevard, Pasadena, CA91125, U.S.A. e-mail: dconlon@caltech.edu
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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 × n matrix with all entries 0 or 1 containing no s × 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.

MSC classification

Primary: 05C35: Extremal problems
Secondary: 05D40: Probabilistic methods
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 173 , Issue 1 , July 2022 , pp. 155 - 161
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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