5 Maximal Sidon Sets and Matroids

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. 1. An extremal problem for Sidon sets Let A be a subset of an abelian group Γ. Two h-tuples (a1, . . . , ah) and (a ′ 1, . . . , a ′ h) of elements of A are called equivalent, denoted (a1, . . . , ah) ∼ (a ′ 1, . . . , a ′ h), if there is a permutation σ of the set {1, . . . , h} such that a′i = aσ(i) for i = 1, . . . , h. If the h-tuples (a1, . . . , ah) and (a ′ 1, . . . , a ′ h) are equivalent, then a1 + · · · + ah = a′1 + · · ·+ a ′ h. We write (a1, . . . , ah) 6∼ (a ′ 1, . . . , a ′ h) if the h-tuples (a1, . . . , ah) and (a ′ 1, . . . , a ′ h) are not equivalent. The h-fold sumset of A, denoted hA, is the set of all elements of Γ that can be written as the sum of h elements of A, with repetitions allowed. For every x ∈ Γ, the representation function rA,h(x) counts the number of inequivalent representations of x as a sum of h elements of A, that is, the number of equivalence classes of h-tuples (a1, . . . , ah) such that x = a1 + · · ·+ ah. The set A is called a Sidon set of order h or a Bh-set if every element of the sumset hA has a unique representation as the sum of h elements of A, that is, if rA,h(x) = 1 for all x ∈ hA. This means that if a1, . . . , ah, a′1, . . . , a ′ h ∈ A and a1 + · · ·+ ah = a ′ 1 + · · ·+ a ′ h, then (a1, . . . , ah) ∼ (a′1, . . . , a ′ h). A Sidon set is a Sidon set of order 2. Let X be a subset of the group Γ, and denote by Bh(X) the set of all finite Bh-sets contained in X . Every set is a B1-set, and Bh(X) ⊆ Bh−1(X) ⊆ · · · ⊆ B2(X) ⊆ B1(X). Moreover, {a} ∈ Bh(X) for all a ∈ X and h ≥ 1. 2000 Mathematics Subject Classification. 11B34,11B75,05B35,05A17,05B40.