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  "body": "\n## Statement\n\nLet $n \\ge r \\ge 1$ be integers, and let $G$ be a simple [[graph|0JD5C82QMB]] with $n$ vertices that has no [[clique|0G52BPNPKG]] with $r + 1$ vertices.\nThen $G$ has at most as many edges as the [[Turán graph|0VTCABADBE]] $T(n, r)$.\nIn particular, $G$ has at most $\\left(1 - \\frac{1}{r}\\right) \\frac{n^2}{2}$ edges.\n\n## Remarks\n\nThe bound is sharp, because $T(n, r)$ has no clique with $r + 1$ vertices.\nThe case $r = 2$ is [[Mantel's theorem|0MXEHSCYCS]] on graphs without triangles.\nThe theorem is a central result of extremal graph theory.\n\n## History\n\nPál Turán described the theorem and the Turán graphs in 1941.\n\n## Explanations\n\n### L1 human\n\nHow many edges can a graph have if no $r + 1$ of its vertices are all joined to each other?\nThe answer is the Turán graph: split the vertices into $r$ balanced teams and join all pairs across teams.\n",
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