Zero-sum problem

In number theory, zero-sum problems are certain kinds of combinatorial problems about the structure of a finite abelian group. Concretely, given a finite abelian group G and a positive integer n, one asks for the smallest value of k such that every sequence of elements of G of size k contains n terms that sum to 0.

The classic result in this area is the 1961 theorem of Paul Erdős, Abraham Ginzburg, and Abraham Ziv.^[1] They proved that for the group ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ of integers modulo n,

$${\displaystyle k=2n-1.}$$

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, but that the same is not true of multisets of size 2n − 2. (Indeed, the lower bound is easy to see: the multiset containing n − 1 copies of 0 and n − 1 copies of 1 contains no n-subset summing to a multiple of n.) This result is known as the Erdős–Ginzburg–Ziv theorem after its discoverers. It may also be deduced from the Cauchy–Davenport theorem.^[2]

More general results than this theorem exist, such as Olson's theorem, Kemnitz's conjecture (proved by Christian Reiher in 2003^[3]), and the weighted EGZ theorem (proved by David J. Grynkiewicz in 2005^[4]).