Transverse Families of Matchings in the Plane

We investigate the following problem: given a set \(S \subset \mathbb{R}^2\) in general position and a positive integer \(k\), find a family of matchings \(\{M_1, M_2, \ldots, M_k\}\) determined by \(S\) such that if \(i \neq j\) then each segment in \(M_i\) crosses each segment in \(M_j\). We give improved linear lower bounds on the size of the matchings in such a family.