An upper bound on the size of separated matchings

We consider a 2n-element equicolored point set (that is, the half of the points will be red and the other half blue) in the plane in convex position. Edges are straight line segments connecting points of different color. A separated matching is a geometrically non-crossing matching where all edges can be crossed by a line. The size of a separated matching is the number of points in it. e show several configurations allowing separated matchings of size at most 4 3 n + O ( n ) . Among these configurations we present a class. This result draws attention to the separated matching conjecture [Hajnal, P., and V. Mészáros, Note on noncrossing alternating path in colored convex sets, accepted to Discrete Mathematics and Theoretical Computer Science], [Kynčl, J., J. Pach and G. Tóth, Long alternating paths in bicolored point sets, Discrete Mathematics 308 19 (2008), 4315–4321].