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  "body": "\n## Problem\n\nLet $m, n, s, t$ be positive integers with $s \\le m$ and $t \\le n$.\nCall an $m \\times n$ matrix with entries in $\\set{0, 1}$ *$(s, t)$-free* if no $s$ rows and $t$ columns cross only in entries equal to $1$.\nLet $z(m, n; s, t)$ be the largest number of entries equal to $1$ in an $(s, t)$-free $m \\times n$ matrix.\nDetermine $z(m, n; s, t)$, or its order of growth.\n\n## Remarks\n\nEquivalently, consider [[bipartite graphs|0ZVQKTRKVQ]] with parts $A$ and $B$ of sizes $m$ and $n$.\nThen $z(m, n; s, t)$ is the largest number of edges of such a graph with no [[complete bipartite|0V0GC7BQAN]] subgraph $K_{s,t}$ that has $s$ vertices in $A$ and $t$ in $B$.\nIt is closely tied to the [[extremal number|DRGS5TT784]] $\\mathrm{ex}(n, K_{s,t})$.\n\nThe Kővári–Sós–Turán theorem gives the upper bound $z(n, n; t, t) = O(n^{2 - 1/t})$ for fixed $t$.\nFor $s = t = 2$ the order $n^{3/2}$ is correct.\nIf $q$ is a prime power and $n = q^2 + q + 1$, then $z(n, n; 2, 2) = (q + 1)(q^2 + q + 1)$, and the incidence matrix of a [[finite projective plane|2HWB7P1DZN]] of order $q$ attains this value.\nThe order $n^{2 - 1/t}$ is also known to be correct for $t = 3$ and, for $z(n, n; s, t)$, when $s$ is large enough compared with $t$, by algebraic constructions called norm graphs.\nIn general the problem is open; for example, the order of growth of $z(n, n; 4, 4)$ is not known.\n\n## Explanations\n\n### L1 human\n\nFill a square grid with as many ones as you can, but never let $s$ rows and $t$ columns cross in ones only.\nHow many ones can you place?\nFor $2$ rows and $2$ columns the forbidden pattern is a rectangle with ones at its four corners, and the answer grows like $n^{3/2}$.\nFor larger forbidden blocks the exact growth rate is one of the main open questions of extremal graph theory.\n",
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