For large n, the minimum number of monochromatic crossings in any 2-colored straight-line drawing of K_n is Θ(n^4), between 1/33 and 0.11798016 times C(n,4).
Computational Geometry: Theory and Applications 36(1), 2–15 (2006)
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On the 2-colored crossing number
For large n, the minimum number of monochromatic crossings in any 2-colored straight-line drawing of K_n is Θ(n^4), between 1/33 and 0.11798016 times C(n,4).