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Maximal Sidon Sets and Matroids

J. A. Dias da Silva·Melvyn B. Nathanson
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Abstract

Let X be a subset of an abelian group and a_1,...,a_h,a'_1,...,a'_h a sequence of 2h elements of X such that a_1 + ... + a_h = a'_1 + ... + a'_h. The set X is a Sidon set of order h if, after renumbering, a_i = a'_i for i = 1,..., h. For k \leq h, the set X is a generalized Sidon set of order (h,k), if, after renumbering, a_i = a'_i for i = 1,..., k. It is proved that if X is a generalized Sidon set of order (2h-1,h-1), then the maximal Sidon sets of order h contained in X have the same cardinality. Moreover, X is a matroid where the independent subsets of X are the Sidon sets of order h.

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J. A. Dias da Silva, Melvyn B. Nathanson. 2005-04-11. Maximal Sidon Sets and Matroids. https://arxiv.org/abs/math/0504226

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