Maximal Sidon Sets and Matroids

Original · EN

Let X be a subset of an abelian group and a₁,...,aₕ,a'₁,...,a'ₕ a sequence of 2h elements of X such that a₁ +... + aₕ = a'₁ +... + a'ₕ. The set X is a Sidon set of order h if, after renumbering, aᵢ = a'ᵢ for i = 1,..., h. For k ≤ h, the set X is a generalized Sidon set of order (h,k), if, after renumbering, aᵢ = 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.