关于Zarankiewicz问题的几点注记
Some remarks on the Zarankiewicz problem
摘要：
Zarankiewicz问题要求对$z(m, n; s, t)$进行估计，即在所有条目$0$或$1$不包含完全由$1$组成的$s \times t$子矩阵的$m \times n$矩阵中$1$的最大数目。我们表明，一个经典的上界$z(m, n; s, t)$由于Kvári，Sós和Turán是紧到常数的参数范围很广。证明依赖于随机代数方法的一个新的定量变量。
英文摘要：
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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