Given a finite subset A of an abelian group G, we study the set k â§ A of all sums of k distinct elements of A. In this paper, we prove that |k â§ A| â¥ |A| for all k ∈ {2,.i¾ .i¾ .,|A|-2}, unless k ∈ {2, |A|-2} and A is a coset of an elementary 2-subgroup of G. Furthermore, we characterize those finite sets A â G for which |k â§ A| = |A| for some k ∈ {2,.i¾ .i¾ .,|A|-2}. This result answers a question of Diderrich. Our proof relies on an elementary property of proper edge-colourings of the complete graph.
['Benjamin Girard', 'Simon Griffiths', 'Yahya Ould Hamidoune']
K-sums in abelian groups