Maximal Sidon sets and matroids

A subset X of an abelian group Γ , written additively, is a Sidon set of order h 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 ∈ I m i a i = ∑ j ∈ J n j b 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 ( 2 h − 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 .