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Published in:
2013 | OriginalPaper | Chapter
The Number of Homothetic Subsets
Authors : Miklós Laczkovich, Imre Z. Ruzsa
Published in: The Mathematics of Paul Erdős I
Publisher: Springer New York
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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.