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  "body": "\n## Statement\n\nLet $G$ be a finite simple [[graph|00C0BR7D7H]] with $n$ vertices and $e$ edges, and let $\\operatorname{cr}(G)$ be its [[crossing number|SWJZ4QRF9W]].\nIf $e \\ge 4n$, then\n$$\\operatorname{cr}(G) \\ge \\frac{e^3}{64 n^2}.$$\n\n## Remarks\n\nA planar simple graph with $n \\ge 3$ vertices has at most $3n - 6$ edges by the [[edge bound for planar graphs|0SM1REAVYJ]].\nRemoving one edge per crossing from a drawing gives a planar graph, so $\\operatorname{cr}(G) \\ge e - 3n$ for every simple graph.\nNow keep each vertex of an optimal drawing independently with probability $p$, as in the [[probabilistic method|3B55HNS754]].\nTaking expectations in the weak bound gives $p^4 \\operatorname{cr}(G) \\ge p^2 e - 3pn$, and the choice $p = 4n/e$ gives the result.\nExample: the complete graph $K_n$ has $e = n(n-1)/2$ edges, which is at least $4n$ for $n \\ge 9$, so $\\operatorname{cr}(K_n) \\ge c n^4$ for a constant $c \u003e 0$.\nWithout a hypothesis like $e \\ge 4n$ the bound fails: a graph with $e = n$ edges can be a cycle, which has crossing number $0$.\n\n## Explanations\n\n### L1 human\n\nWhen a graph has many more edges than vertices, any drawing of it in the plane must have many crossings.\nThe inequality says how many: at least a constant times $e^3 / n^2$.\nDoubling the number of edges, with the same vertices, forces about eight times as many crossings.\n\n### L2 human\n\nThe inequality was proved by Ajtai, Chvátal, Newborn and Szemerédi and, independently, by Leighton, in the early 1980s.\nIts main use in discrete geometry is through incidence bounds.\nLászló Székely used it to give short proofs of the [[Szemerédi–Trotter theorem|NQBWFSC1E0]] on point-line incidences and of the $O(n^{4/3})$ bound in the [[unit distance problem|EPWB2JEXPE]].\nOne draws the points as vertices and the pieces of lines or circles as edges, and compares the forced crossings with the number of pairs of curves.\n",
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