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  "body": "\n## Summary\n\nExtremal graph theory studies how large or small a [[graph|00C0BR7D7H]] can be when it must have, or must avoid, a given structure.\nIts basic quantity is the extremal number $\\mathrm{ex}(n, H)$, the largest number of edges of a graph on $n$ vertices with no [[subgraph|0WBFFC8XZ9]] isomorphic to $H$.\n[[Turán's theorem|0JG1KZSPFZ]] computes it for complete graphs, and the [[Erdős–Stone theorem|XZF9BFJFXB]] gives it asymptotically for every non-bipartite $H$.\nFor bipartite $H$ the question leads to the [[Zarankiewicz problem|KT9WJFRQ3F]], which is still largely open.\n\n## Explanations\n\n### L1 human\n\nExtremal graph theory asks questions of the form \"how many edges can you draw before a certain pattern is forced to appear?\".\nFor example, a graph on $n$ vertices without a triangle has at most $n^2/4$ edges.\nThe best graph puts the vertices in two equal groups and joins every vertex of one group to every vertex of the other.\nResults of this kind show that density alone forces structure.\n",
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