Maximal Sidon sets and matroids

A subset X of an abelian group Γ, written additively, is a Sidon set of order h if whenever {(ai,mi):i∈I} and {(bj,nj):j∈J} are multisets of size h with elements in X and ∑i∈Imiai=∑j∈Jnjbj, then {(ai,mi):i∈I}={(bj,nj):j∈J}. The set X is a generalized Sidon set of order (h,k) if whenever two such multisets have the same sum, then their multiset intersection has size at least 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.