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There's a well-established result which provides a lower bound for the crossing number of any simple undirected graph. However, is there any known result for an upper bound in this setting?

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If the graph has $m$ edges then $\binom{m}{2}$ is tight. You could object that you want to use straight lines, but that is not a requirement for the lower bound version of the problem. –  Chris Godsil Mar 14 '13 at 12:06

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