Consider a
k-element subset
P of the plane. It is known that the maximum number of sets similar to
P that can be found among
n points in the plane is
(
n
2) if and only if the cross ratio of any quadruplet of points in P is algebraic [3], [9].
In this paper we study the structure of the extremal n-sets A which have cn
2 similar copies of P. As our main result we prove the existence of large lattice-like structures in such sets A. In particular we prove that, for n large enough, A must contain m points in a line forming an arithmetic progression, or m × m lattices, when P is not cocyclic or collinear. On the other hand we show that for cocyclic or collinear sets P, there are n-element sets A with c
P
n
2 copies of P and without k × k lattice subsets.
Mathematics Subject Classification (2000):
52C10
* Supported by Hungarian National Foundation for Scientific Research (OTKA), Grants No. T 42750 and T 47056.