 
Summary: JOURNAL OF COMBINATORIAL THEORY, Series A 41, 154157 (1986)
Note
The Number of Small Semispaces
of a Finite Set of Points in the Plane
N~CA ALON*
Department of Mathematics, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139
AND
E. GY~RI
Mathematical Institute of the Hungarian
Academy of Sciences, Budapest, Hungary
Communicated by the Managing Editors
Received May 22, 1984
For a configuration S of n points in the plane, let gk(S) denote the number of
subsets of cardinality
man and Pollack (J. Combin. Theory Ser. A 36 (1984), 101104) showed that if
k < n/2 then g,,, < 2nk  2k2 k. Here we show that g,," = k. n for k
Academic Press, Inc.
Let S be a finite set of points in the plane. Following Goodman and
Pollack [GP2] we call the intersection of S with a half plane a semispace
