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  "body": "\n## Statement\n\nFor integers $r \\ge 1$ and $t \\ge 1$, let $K_{r+1}(t)$ be the complete $(r+1)$-partite [[graph|00C0BR7D7H]] with $r + 1$ parts of $t$ vertices each.\nLet $r \\ge 1$ and $t \\ge 1$ be integers and let $\\varepsilon \u003e 0$ be a real number.\nThen there is an integer $n_0$ such that every graph with $n \\ge n_0$ vertices and at least $\\left(1 - \\frac{1}{r} + \\varepsilon\\right) \\frac{n^2}{2}$ edges contains a [[subgraph|0WBFFC8XZ9]] isomorphic to $K_{r+1}(t)$.\n\n## Remarks\n\nThe main consequence is a formula for the [[extremal number|DRGS5TT784]].\nLet $H$ be a graph with [[chromatic number|0VHJRXNAAS]] $\\chi(H) = r + 1 \\ge 2$.\nThen $H$ is a subgraph of $K_{r+1}(t)$ for $t = |V(H)|$, and the [[Turán graph|0VTCABADBE]] $T(n, r)$ is $H$-free, so\n$$\\mathrm{ex}(n, H) = \\left(1 - \\frac{1}{r}\\right) \\frac{n^2}{2} + o(n^2) \\quad (n \\to \\infty).$$\nFor example, the Petersen graph has chromatic number $3$, so a graph on $n$ vertices with no Petersen subgraph has at most $n^2/4 + o(n^2)$ edges.\nFor bipartite $H$ ($r = 1$) the formula only says $\\mathrm{ex}(n, H) = o(n^2)$; the true order is the subject of the [[Zarankiewicz problem|KT9WJFRQ3F]].\nA modern proof applies the [[Szemerédi regularity lemma|ETMAPNQY7T]] to find a dense $K_{r+1}$ in the reduced graph, and then a counting lemma to blow it up to $K_{r+1}(t)$.\n\n## Explanations\n\n### L1 human\n\n[[Turán's theorem|0JG1KZSPFZ]] says how many edges force a complete graph $K_{r+1}$.\nThe Erdős–Stone theorem says that only a tiny bit more forces a much larger graph: a \"blown-up\" $K_{r+1}$ with many vertices in each part.\nAs a result, the largest number of edges avoiding any fixed graph $H$ is governed, up to a small error, by its chromatic number alone.\n\n### L2 human\n\nThe theorem is often called the fundamental theorem of [[extremal graph theory|SC5QB5F4TE]].\nIt gives $\\mathrm{ex}(n, H) / \\binom{n}{2} \\to 1 - \\frac{1}{\\chi(H) - 1}$ for every graph $H$ with at least one edge.\nIt is sharp for non-bipartite $H$ and degenerate for bipartite $H$, where the extremal number is $o(n^2)$.\nStability results refine it: an $H$-free graph with nearly $\\mathrm{ex}(n, H)$ edges is close in structure to the Turán graph.\n",
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