In number theory, zero-sum problems are a certain class of combinatorial questions. In general, a finite abelian group G is considered. The zero-sum problem for the integer n is the following: Find the smallest integer k such that every sequence of elements of G with length contains n terms that sum to 0.
Explicitly this says that any multiset of 2n − 1 integers has a subset of size n the sum of whose elements is a multiple of n. This result is known as the Erdős–Ginzburg–Ziv theorem after its discoverers: it may be deduced from the Cauchy–Davenport theorem.
More general results than this theorem exist, such as Olson's theorem, Kemnitz's conjecture (proved by Christian Reiher in 2003), and the weighted EGZ theorem (proved by David J. Grynkiewicz in 2005).
- Erdős, Paul; Ginzburg, A.; Ziv, A. (1961). "A theorem in additive number theory". Bull. Res. Council Israel 10F: 41–43. Zbl 0063.00009.
- Nathanson (1996) p.48
- Reiher, Christian (2007), "On Kemnitz' conjecture concerning lattice-points in the plane", The Ramanujan Journal 13 (1–3): 333–337, doi:10.1007/s11139-006-0256-y, Zbl 1126.11011.
- Grynkiewicz, D. J. (2006), "A Weighted Erdős-Ginzburg-Ziv Theorem", Combinatorica 26 (4): 445–453, doi:10.1007/s00493-006-0025-y, Zbl 1121.11018.
- Geroldinger, Alfred (2009). "Additive group theory and non-unique factorizations". In Geroldinger, Alfred; Ruzsa, Imre Z. Combinatorial number theory and additive group theory. Advanced Courses in Mathematics CRM Barcelona. Elsholtz, C.; Freiman, G.; Hamidoune, Y. O.; Hegyvári, N.; Károlyi, G.; Nathanson, M.; Solymosi, J.; Stanchescu, Y. With a foreword by Javier Cilleruelo, Marc Noy and Oriol Serra (Coordinators of the DocCourse). Basel: Birkhäuser. pp. 1–86. ISBN 978-3-7643-8961-1. Zbl 1221.20045.
- Nathanson, Melvyn B. (1996). Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Graduate Texts in Mathematics 165. Springer-Verlag. ISBN 0-387-94655-1. Zbl 0859.11003.