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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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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DOI: https://doi.org/10.1007/978-1-4614-7258-2_32
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