Maximal Sidon Sets and Matroids

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.