  Let $D$ be a straight-line drawing of a graph. The rectilinear 2-colored
crossing number of $D$ is the minimum number of crossings between edges of the
same color, taken over all possible 2-colorings of the edges of $D$. First, we
show lower and upper bounds on the rectilinear 2-colored crossing number for
the complete graph $K_n$. To obtain this result, we prove that asymptotic
bounds can be derived from optimal and near-optimal instances with few
vertices. We obtain such instances using a combination of heuristics and
integer programming. Second, for any fixed drawing of $K_n$, we improve the
bound on the ratio between its rectilinear 2-colored crossing number and its
rectilinear crossing number.
