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Prove that there exists constant k such that, for all $5v < e$ there is a subgraph of the complete graph of $v$ vertics with crossing number less or equal than $ k e^3/v^2$.

Any hints for a way to apply probabilistic argument ?

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For an excellent survey and discussion of issues related to geometric incidences and crossings, look at the paper by Janos Pach and Micha Sharir entitled Geometric Incidences, in the book: Towards a Theory of Geometric Graphs, Volume 342, Contemporary Mathematics, American Mathematical Society, 2004. –  Joseph Malkevitch Feb 6 '12 at 17:03
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You may want to look at crossing number inequality of this link.

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I am familiar with the crossing number theorem. This question is actually an exercise after the theorem. –  ebg9 Dec 8 '11 at 13:23
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