A zero-sum theorem (Q1865416)

For a nontrivial abelian group \(G\) of order \(m\), \(G\)-colorings of the integer set \([1,n]:=\{1,2,\ldots,n\}\) associating one element of \(G\) with every element of \([1,n]\) are considered. An \(m\)-tuple \((x_1,\dots,x_m)\) of integers from \([1,n]\) is called zero-sum with respect to a given coloring if the sum of the images of these integers in \(G\) is zero. It is shown that \(n=m(m-1)+1\) is the smallest integer such that every \(G\)-coloring of \([1,n]\) contains a zero-sum solution to the inequality \(x_1+x_2+\cdots+x_{m-1}<x_m\). For cyclic abelian groups \(G\) this yields an Erdős-Ginzburg-Ziv type generalization of a known result concerning a monochromatic solution of the above inequality in a 2-coloring of the positive integers.