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Problems on Repeated Subconfigurations

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Research Problems in Discrete Geometry

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Abstract

The set of unit-distance pairs determined by an n-element point set X can be regarded as an equivalence class of all point pairs under congruence as the equivalence relation. The basic questions discussed in Chapter 5 are to determine the size of the largest equivalence class and the number of distinct equivalence classes. Erdős and Purdy [ErP71], [ErP76] started the investigation of the same questions for k-dimensional simplices in IRd, that is, for (k + 1)-tuples rather than point pairs. (In [ErP71] some of these problems are attributed to A. Oppenheim.) Let u k,d(n) denote the maximum number of mutually congruent k-dimensional simplices determined by n points in d-dimensional Euclidean space. Using the notation in the previous chapter, u 1,d (n) = u d(n) is the maximum number of unit distances determined by n points in IRd.

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(2005). Problems on Repeated Subconfigurations. In: Research Problems in Discrete Geometry. Springer, New York, NY. https://doi.org/10.1007/0-387-29929-7_7

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