Non-canonical Extensions of Erdős-Ginzburg-Ziv Theorem

In 1961, Erdös-Ginzburg-Ziv proved that for a given natural number n ≥ 1 and a sequence a₁, a₂, ··· , a_2n−1 of integers (not necessarily distinct), there exist 1 ≤ i₁ < i₂ < ··· < i_n ≤ 2n − 1 such that a_i1 + a_i2 + ··· + a_in is divisible by n. Moreover, the constant 2n − 1 is tight. By now, there are many canonical generalizations of this theorem. In this paper, we shall prove some non-canonical generalizations of this theorem.