k-Sums in Abelian Groups

BENJAMIN GIRARD, SIMON GRIFFITHS, YAHYA OULD HAMIDOUNE
2012 Combinatorics, probability & computing  
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 in 2,...,|A|-2, unless k is in 2,|A|-2 and A is a coset of an elementary 2-subgroup of G. Furthermore, we characterize those finite subsets A of G for which |k ∧ A| = |A| for some k in 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.
doi:10.1017/s0963548312000168 fatcat:3ymk6axrnfeqtj3voymoholznu