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POINT-SETS WITH MANY SIMILAR COPIES OF A PATTERN: WITHOUT A FIXED NUMBER OF
 

Summary: POINT-SETS WITH MANY SIMILAR COPIES OF A
PATTERN: WITHOUT A FIXED NUMBER OF
COLLINEAR POINTS OR
IN PARALLELOGRAM-FREE POSITION
BERNARDO M. ŽABREGO AND SILVIA FERNŽANDEZ-MERCHANT
Department of Mathematics
California State University, Northridge
18111 Nordhoff St, Northridge, CA 91330-8313.
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 parallelogram-free 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

  

Source: Abrego, Bernardo - Department of Mathematics, California State University, Northridge
Fernandez, Silvia - Department of Mathematics, California State University, Northridge

 

Collections: Mathematics