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 Θ(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.

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Correspondence to Bernardo M. Ábrego.

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* Supported by Hungarian National Foundation for Scientific Research (OTKA), Grants No. T 42750 and T 47056.

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Ábrego, B.M., Elekes*, G. & Fernández-Merchant, S. Structural Results For Planar Sets With Many Similar Subsets. Combinatorica 24, 541–554 (2004). https://doi.org/10.1007/s00493-004-0033-8

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Mathematics Subject Classification (2000):

  • 52C10