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I find out that this problem is a special case of Zarankiewicz problem and it was solved by BollobasIstván Reiman in 20041958 (thanks to Oliver Krüger for correction). The answer is $ex(n,n,C_4)\sim n^{3/2}$.
I find out that this problem is a special case of Zarankiewicz problem and it was solved by Bollobas in 2004. The answer is $ex(n,n,C_4)\sim n^{3/2}$.
I find out that this problem is a special case of Zarankiewicz problem and it was solved by István Reiman in 1958 (thanks to Oliver Krüger for correction). The answer is $ex(n,n,C_4)\sim n^{3/2}$.
