Perfect Matchings with Crossings

For sets of n points, n even, in general position in the plane, we consider straight-line drawings of perfect matchings on them. It is well known that such sets admit at least $C_{n/2}$ different plane perfect matchings, where $C_{n/2}$ is the n/2-th Catalan number. Generalizing this result we are interested in the number of drawings of perfect matchings which have k crossings. We show the following results. (1) For every $k\le \frac{1}{64}{n}^2-\frac{35}{32}n\sqrt{n}+\frac{1225}{64}n$, any set with n points, n sufficiently large, admits a perfect matching with exactly k crossings. (2) There exist sets of n points where every perfect matching has at most $\frac{5}{72}{n}^2-\frac{n}{4}$ crossings. (3) The number of perfect matchings with at most k crossings is superexponential in n if k is superlinear in n. (4) Point sets in convex position minimize the number of perfect matchings with at most k crossings for $k=0,1,2$, and maximize the number of perfect matchings with $\left(\begin{array}{c}n/2\\ 2\end{array}\right)$ crossings and with $\left(\begin{array}{c}n/2\\ 2\end{array}\right)-1$ crossings.