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  "body": "\n## Definition\n\nLet $H$ be a finite simple [[graph|00C0BR7D7H]] with at least one edge, and let $n \\ge 1$ be an integer.\nA graph $G$ is *$H$-free* if no [[subgraph|0WBFFC8XZ9]] of $G$ is [[isomorphic|0F7QKGJS6B]] to $H$.\nThe *extremal number* $\\mathrm{ex}(n, H)$ is the largest number of edges of an $H$-free graph with $n$ vertices.\nAn $H$-free graph with $n$ vertices and $\\mathrm{ex}(n, H)$ edges is an *extremal graph* for $H$.\nExtremal graph theory is the study of $\\mathrm{ex}(n, H)$, of the extremal graphs, and of similar maximum and minimum problems for graphs.\n\n## Remarks\n\nExample: $\\mathrm{ex}(4, K_3) = 4$.\nThe $4$-cycle has $4$ edges and no triangle.\nA graph on $4$ vertices with $5$ edges is $K_4$ minus one edge, and it contains a triangle.\n\nNon-example: $K_4$ is not counted for $\\mathrm{ex}(4, K_3)$, because it is not $K_3$-free.\n\nThe maximum exists, since the empty graph on $n$ vertices is $H$-free and the number of edges is at most $\\binom{n}{2}$.\nFor complete graphs, [[Turán's theorem|0JG1KZSPFZ]] says that $\\mathrm{ex}(n, K_{r+1})$ is the number of edges of the [[Turán graph|0VTCABADBE]] $T(n, r)$.\n",
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