On a zero-sum generalization of a variation of Schur's equation (Q1015440)

In this paper the authors study a variant of Schur's problem: Let \(m\geq 3\) be a positive integer. Let \(R(L_m;2)\) (or \(R(L_m; \mathbb{Z}_m\)) respectively) denote the minimum integer \(N\) such that for every function \(\Delta:\{1,2, \ldots , N\} \rightarrow \{0,1\}\) (or \(\Delta:\{1,2, \ldots , N\} \rightarrow \mathbb{Z}_m\)) there exist \(m\) integers \(x_1<x_2 < \cdots < x_m\) with \(\sum_{i=1}^{m-1} x_i <x_m\) and \(\Delta(x_1)=\Delta(x_2)= \cdots =\Delta(x_m)\) (and \(\sum_{i=1}^m \Delta(x_i)=0\)). In this paper it is proved that \(R(L_m;2)=R(L_m;\mathbb{Z}_m)\), for every odd prime \(m\). An explicit value of \(R(L_m;2)\) had been worked out in [\textit{A. Bialostocki} and \textit{D. Schaal}, ''On a variation of Schur numbers,'' Graphs Comb. 16, No.\,2, 139-147 (2000; Zbl 0973.05080)].