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The Mathematics of Paul Erdős I pp 523–532Cite as

The Number of Homothetic Subsets

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Summary

We investigate the maximal number S(P, n) of subsets of a set of n elements homothetic to a fixed set P. Elekes and Erdős proved that S(P, n) > cn 2 if | P | = 3 or the elements of P are algebraic. For | P | ≥ 4 we show that S(P, n) > cn 2 if and only if every quadruple in P has an algebraic cross ratio. Moreover, there is a sequence S n of numbers such that \(S(P,n) \asymp S_{n}\) whenever | P | = 4 and the cross ratio of P is transcendental.

Supported by Hungarian National Foundation for Scientific Research, Grant T 7582.

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References

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Author information

Authors and Affiliations

  1. Department of Analysis, Eötvös Loránd University, Múzeum krt. 6–8, Budapest, H-I088, Hungary

    Miklós Laczkovich

  2. Department of Mathematics, University College London, London, WC1E 6BT, UK

    Miklós Laczkovich

  3. Mathematical Institute of the Hungarian Academy of Sciences, Budapest, 127, H-1364, Hungary

    Imre Z. Ruzsa

Authors
  1. Miklós Laczkovich
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  2. Imre Z. Ruzsa
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Correspondence to Miklós Laczkovich .

Editor information

Editors and Affiliations

  1. Dept. Comp. Sci. & Engineering, University of California, San Diego, La Jolla, California, USA

    Ronald L. Graham

  2. Department of Applied Mathematics, Charles University Department of Applied Mathematics, Prague, Czech Republic

    Jaroslav Nešetřil

  3. Department of Mathematics, Iowa State University, Ames, Iowa, USA

    Steve Butler

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Laczkovich, M., Ruzsa, I.Z. (2013). The Number of Homothetic Subsets. In: Graham, R., Nešetřil, J., Butler, S. (eds) The Mathematics of Paul Erdős I. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7258-2_32

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