Perfect Matchings with Crossings

In this paper, we analyze the number of straight-line perfect matchings with $k$ crossings on point sets of size $n$ = $2m$ in general position. We show that for every $kn/8-1)$, every $n$-point set admits a perfect matching with exactly $k$ crossings and that there exist $n$-point sets where every perfect matching has fewer than $5n^2/72$ crossings. We also study the number of perfect matchings with at most $k$ crossings. Finally we show that convex point sets %in convex position maximize the number of perfect matchings with $n/2 $ crossings and $n/2 -1$ crossings.