 
Summary: Disjoint Simplices and Geometric
Hypergraphs
J. AKIYAMA~ANDN. ALON~
a Department of Mathematics
Tokai University
Hiratsuka 25912, Japan
Department of Mathematics
Tel Aviv University
69978 Tel Aviv, Israel
and
Bell CommunicationsResearch
Morristown, New Jersey 07960
INTRODUCTION
Let A be a set of 2n points in general position in the Euclidean plane R2,and
suppose n of the points are colored red and the remaining n are colored blue. A
celebrated Putnam problem (see [6]) asserts that there are n pairwise disjoint
straight line segments matching the red points to the blue points. To show this,
consider the set of all n! possible matchings and choose one, M, that minimizes the
sum of lengths I(M)of its line segments. It is easy to show that these line segments
cannot intersect.Indeed, if the two segmentsv,, b, and v2,b, intersect, where v,, v2
