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

## TL;DR

This paper improves the upper bound on crossing-free matchings by refining the recurrence relation used in previous bounds, leading to tighter estimates in combinatorial geometry.

## Contribution

The authors strengthen the recurrence relation in Sharir and Welzl's analysis, resulting in improved bounds on crossing-free matchings.

## Key findings

- Enhanced recurrence relations for crossing-free matchings
- Tighter upper bounds on the number of crossing-free matchings
- Improved theoretical understanding of geometric matchings

## Abstract

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.