A survey on zero-sum problems

Abstract

In 1961, Erd\"{o}s, Ginzburg and Ziv proved, so called the EGZ theorem, a simple but important theorem. The theorem states that for any positive integer $n$, every sequence $a_1,a_2,\ldots, a_{2n-1}$ of integers has a subsequence $a_{i_1},a_{i_2},\ldots,a_{i_n}$ such that $a_{i_1}+a_{i_2}+\cdots+a_{i_n}$ is divisible by $n$. In this thesis, we survey various results related with the EGZ theorem. In particular, in Section 4, we introduce Bialostocki's conjecture, which is one of generalizations of the EGZ theorem. We consider several particular cases of Bialostocki's conjecture and give a proof of these cases. We also explain some relations between some well known theorems and Bialostocki's conjecture.