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Journal Articles Combinatorics, Probability and Computing Year : 2012

k-sums in abelian groups

Abstract

Given a finite subset A of an abelian group G, we study the set k \wedge A of all sums of k distinct elements of A. In this paper, we prove that |k \wedge 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 \wedge 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.
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Dates and versions

hal-00630441 , version 1 (10-10-2011)
hal-00630441 , version 2 (26-06-2012)

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Benjamin Girard, Simon Griffiths, Yahya Ould Hamidoune. k-sums in abelian groups. Combinatorics, Probability and Computing, 2012, 21 (4), pp.582-596. ⟨10.1017/S0963548312000168⟩. ⟨hal-00630441v2⟩
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