Maximal Sidon sets and matroids

A subset X of an abelian group @C, written additively, is a Sidon set of orderh if whenever {(a\"i,m\"i):i@?I} and {(b\"j,n\"j):j@?J} are multisets of size h with elements in X and @?\"i\"@?\"Im\"ia\"i=@?\"j\"@?\"Jn\"jb\"j, then {(a\"i,m\"i):i@?I}={(b\"j,n\"j):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.