 
Summary: Discrete ComputGeom4:287290(1989)
tG989Discrete&Cq~nputati,.ml
eomet
Disjoint Edges in Geometric Graphs
N. Alont* and P. ErdSs2
l Departmentof Mathematics,Sackler Facultyof ExactSciences,
Tel AvivUniversity,RamatAviv,Tel Aviv,Israel
2Mathematical Instituteof the HungarianAcademyof Sciences,
Budapest, Hungary
Abstract. Answering an old question in combinatorial geometry, we show that any
configuration consisting of a set V of n points in general position in the plane and
a set of 6n  5 closed straight line segments whose endpoints lie in V,contains three
pairwise disjoint line segments.
A geometric graph is a pair G = (V, E), where V is a set of points (=vertices) in
general position in the plane, i.e., no three on a line, and E is a set of distinct,
closed, straight line segments, called edges, whose endpoints lie in V. An old
theorem of the second author [Er] (see also [Ku] for another proof), states that
any geometric graph with n points and n + 1 edges contains two disjoint edges,
and this is best possible for every n >3. For k >2, let f(k n) denote the maximum
number of edges of a geometric graph on n vertices that contains no k pairwise
