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Sharper Zarankiewicz and Diagonal Bipartite Ramsey Bounds
We prove that there is an absolute positive constant $c$ such that every bipartite graph with $N$ vertices in each part and at least $N^2/2$ edges contains a complete bipartite graph $K_{t,t}$ whenever $N\ge c\, 2^{t}$. This improves the classical Kővári-Sós-Turán bound requiring $N$ of order $t
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