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Claim. $\mathrm{ex}(n;K_{2,2})\gg n^{3/2}$, the statement of Problem 714 for $r=2$. T. Kővári, V. T. Sós and P. Turán, On a problem of K. Zarankiewicz, Colloq. Math. 3 (1954), 50--57 (the volume carries the year only, so this page's name uses its first day); library source card. With $k_j(n)$ the least number of $1$'s in an $n\times n$ $0$--$1$ matrix that forces a $j\times j$ minor of $1$'s, the paper's (1.3) states $\lim k_2(n)/n^{3/2}=1$; the lower bound is the construction of Section 5 (pp. 54--55), which for $n=p^2$, $p$ prime, takes the $p^2$ sets $J_{ab}=\{kp+\langle a+bk\rangle+1:k=0,\ldots,p-1\}$ of $\{1,\ldots,p^2\}$, any two sharing at most one element, as the rows of a matrix with $p^3=n^{3/2}$ ones and no $2\times2$ minor of $1$'s, and the passage to all $n$ is the paper's. Such a matrix is the biadjacency matrix of a bipartite graph with $n$ vertices on each side, $n^{3/2}$ edges and no $C_4=K_{2,2}$, so $\mathrm{ex}(2n;K_{2,2})\ge(1-o(1))n^{3/2}$, that is, $\mathrm{ex}(N;K_{2,2})\ge(2^{-3/2}-o(1))N^{3/2}$ for even $N$ and, by monotonicity, $\mathrm{ex}(N;K_{2,2})\gg N^{3/2}$ for all $N$; this translation from the matrix to the graph is the paper's (3.1) read in the other direction and is made here. The same paper's (1.5) is the upper bound $k_j(n)<1+jn+[(j-1)^{1/j}n^{2-1/j}]$ for every $j$, and its Section 6 states the conjecture (6.1) that the exponent is right for every $j$, the matrix form of the problem's question (inequality (6.1)).

Covers. The instance $r=2$ of the statement for every $r\ge2$, with the constant $2^{-3/2}$ rather than the sharp $\tfrac12$ that Erdős, Rényi and Sós and Brown reach for non-bipartite graphs; nothing for any $r\ge3$.

Depends on. Nothing in this wiki: the construction and the asymptotic are the paper's, and the matrix-to-graph step is elementary.

Acceptance. Refereed: Colloquium Mathematicum, a refereed journal. No reviewed evidence is listed: the site labels the problem OPEN, and its commentary names the paper for the upper bound only. This corpus supplies no independent proof review: the statements (1.3), (1.5), (3.1) and (6.1) are checked against the print, and the Section 5 construction's proof is not.