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  "body": "\n## Statement\n\nLet $G$ be a finite simple [[graph|0JD5C82QMB]] with $n$ vertices and no cycle of length $3$.\nThen $G$ has at most $\\lfloor n^2/4 \\rfloor$ edges.\n\n## Remarks\n\nThe bound is sharp: the [[complete bipartite graph|0V0GC7BQAN]] $K_{\\lfloor n/2 \\rfloor, \\lceil n/2 \\rceil}$ has no triangle and has $\\lfloor n^2/4 \\rfloor$ edges.\nThe theorem is the case $r = 2$ of [[Turán's theorem|0JG1KZSPFZ]].\n\n## History\n\nWillem Mantel stated the theorem in 1907.\n\n## Explanations\n\n### L1 human\n\nA graph on $n$ vertices without a triangle can have at most about a quarter of $n^2$ edges.\nSplit the vertices into two equal halves and join each pair across: this graph reaches the bound.\n",
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