Crossing number

The crossing number of a graph is the minimum number of edge-crossing pairs among its plane drawings. For a fixed drawing, deleting at most one edge per crossing pair leaves a planar graph.

If a simple graph has $n$ vertices and $e\ge 4n$ edges, every plane drawing has at leastcrossings. Apply the linear planar bound to the random induced subgraph obtained by retaining vertices independently with probability $4n/e$.