  We study extremal type problem arising from the question: What is the maximum
number of edge-disjoint non-crossing perfect matchings on a set S of 2n points
in the plane such that their union is a triangle-free geometric graph? We
approach this problem by considering four different situations of S. In
particular, in the general position, we obtain (i) a sufficient condition for
the existence of n edge-disjoint non-crossing perfect matchings in the general
position whose union is a maximal triangle-free geometric graph, and (ii) a
lower bound on the number of edge-disjoint non-crossing perfect matchings whose
union is a triangle free geometric graph.
