  We consider bichromatic point sets with $n$ red and $n$ blue points and study
straight-line bichromatic perfect matchings on them. We show that every such
point set in convex position admits a matching with at least
$\frac{3n^2}{8}-\frac{n}{2}+c$ crossings, for some $ -\frac{1}{2} \leq c \leq
\frac{1}{8}$. This bound is tight since for any $k> \frac{3n^2}{8}
-\frac{n}{2}+\frac{1}{8}$ there exist bichromatic point sets that do not admit
any perfect matching with $k$ crossings.
