Result: A conjecture in combinatorial number theory

The authors study the following conjecture proposed by \textit{N. Alon, S. Friedland} and \textit{G. Kalai} [J. Comb. Theory, Ser. B 37, 79--91 (1984; Zbl 0527.05059)]: Given \(m\) \(k\)-dimensional integer vectors \(a_i:= (a_{i1},\cdots,a_{ik})\), where \(i=1,2,\dots,m\), there must be a subset \(I\) of the set \(\{1,2,\dots,m\}\) such that \(\sum_{i\in I} a_i \equiv 0\bmod n\), provided that \(n