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Sharir and Welzl [1] derived a bound on crossing-free matchings primarilybased on solving a recurrence based on the size of the matchings. We show thatthe 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 improvingthe upper bound for crossing-free matchings.

Improving Sharir and Welzl's bound on crossing-free matchings through solving a stronger recurrence