Structural Results For Planar Sets With Many Similar Subsets

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 (n2 ) 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 cn2 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

• Type: Journal
• Year: 2004
• Where: COMBINATORICA
• Authors: Bernardo M. Ábrego, György Elekes, Silvia Fernández-Merchant
• Keywords: COMBINATORICA 2004, Cross Ratio, Large Lattice-like Structures, Maximum Number