Zero-sum problem

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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 any sequence of elements of G with length $k$ contains n terms that sum to 0.

In 1961 Paul Erdős, A. Ginzburg, and A. Ziv proved the general result for $\mathbb{Z}/n\mathbb{Z}$ (the integers mod n) that

k = 2n − 1.

Here, the n is size of G, but generally it is not hold (???).

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 generally known as the EGZ theorem after its discoverers.

More general results than this theorem exist, such as Olson's theorem, Kemnitz's conjecture (proved by C. Reiher in 2003), and the weighted EGZ theorem (proved by D. J. Grynkiewicz in 2005).