Disjoint Edges in Geometric Graphs

A stract. Answering an old question in com inatorial 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 est possi le for every n >_ 3. For k>-2, let f(k n) denote the maximum num er of edges of a geometric graph on n vertices that contains no k pairwise disjoint edges. Thus, the result stated a ove is simply the fact f(2, n) = n for all n >_ 3. Kupitz [Ku] and Perles [Pe] (see also [AA]) raised the pro lem of determining or estimating f(kn) for k>_3. In particular, they asked if f(3, n) O(n). This specific pro lem, of determining or estimating f(3, n), was already mentioned in 1966 y Avital and Hanani [AH], and it seems it was a known pro lem even efore that. In this note we answer this question y proving the following. Theorem 1. For every n ? 1, f(3, n) < 6n-5, i.e ., any geometric graph with n vertices and 6n-5 edges contains three pairwise disjoint edges .