Forcing Disjoint Segments in the Plane

Consider a geometric graph given bynpoints in the plane (in general position) andmline segments, each segment joining a pair of the given points. We show that: ifm≥3n+1then there are three pairwise disjoint segments; ifm≥10n+1then there are four disjoint segments; and ifm≥ckn(logn)rk-4then there arekdisjoint segments.