 
Summary: POINTSETS WITH MANY SIMILAR COPIES OF A
PATTERN: WITHOUT A FIXED NUMBER OF
COLLINEAR POINTS OR
IN PARALLELOGRAMFREE POSITION
BERNARDO M. ŽABREGO AND SILVIA FERNŽANDEZMERCHANT
Department of Mathematics
California State University, Northridge
18111 Nordhoff St, Northridge, CA 913308313.
email:{bernardo.abrego,silvia.fernandez}@csun.edu
Abstract. Let P be a finite pattern, that is, a finite set of
points in the plane. We consider the problem of maximizing
the number of similar copies of P over all sets of n points in the
plane under two general position restrictions: (1) Over all sets
of n points with no m points on a line. We call this maximum
SP (n, m). (2) Over all sets of n points with no collinear triples
and not containing the 4 vertices of any parallelogram. These
sets are called parallelogramfree and the maximum is denoted
by SP (n). We prove that SP (n, m) n2
whenever m(n)
as n and that (n log n) SP (n) O(n3/2
