k-Sums in Abelian Groups
BENJAMIN GIRARD, SIMON GRIFFITHS, Yahya Ould Hamidoune · Combinatorics Probability Computing · 2012
Given a finite subset A of an abelian group G, we study the set k ∧ A of all sums of k distinct elements of A. In this paper, we prove that |k ∧ A| ≥ |A| for all k ∈ {2,. . .,|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 ⊆ G for which |k ∧ A| = |A| for some k ∈ {2,. . .,|A| − 2}. This result answers a question of Diderrich. Our proof relies on an elementary property of proper edge-colourings of the complete graph.