  Sharir and Welzl [1] derived a bound on crossing-free matchings primarily
based on solving a recurrence based on the size of the matchings. We show that
the recurrence given in Lemma 2.3 in Sharir and Welzl can be improve to
$(2n-6s)\textbf{Ma}_{m}(P)\leq\frac{68}{3}(s+2)\textbf{Ma}_{m-1}(P)$ and
$(3n-7s)\textbf{Ma}_{m}(P)\leq44.5(s+2)\textbf{Ma}_{m-1}(P)$, thereby improving
the upper bound for crossing-free matchings.
