Geometric Graphs with Few Disjoint Edges

A geometric graph is a graph drawn in the plane so that the vertices are represented by points in general position, the edges are represented by straight line segments connecting the corresponding points. Improving a result of Pach and Torcsik, we show that a geometric graph on n vertices with no k+1 pairwise disjoint edges has at most k super(3) (n+1) edges. On the other hand, we construct geometric graphs with n vertices and approximately (3/2)(k-1)n edges, containing no k+1 pairwise disjoint edges. We also improve both the lower and upper bounds of Goddard, Katchalski, and Kleitman on the maximum number of edges in a geometric graph with no four pairwise disjoint edges.