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 T\"or\H ocsik, we show that a geometric graph on $n$ vertices with no $k+1$ pairwise disjoint edges has at most $k^3(n+1)$ edges. On the other hand, we construct geometric graphs with $n$ vertices and approximately ${3\over 2}(k-1)n$ 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.
Paper Available at:
ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1998/98-22.ps.gz
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