On Constrained \(2\)-Partitions of Monochromatic Sets and Generalizations in the Sense of Erdós-Ginzburg-Ziv

Abstract

Let \(m \geq 4\) be a positive integer and let \({Z}_m\) denote the cyclic group of residues modulo \(m\). For a system \(L\) of inequalities in \(m\) variables, let \(R(L;2)\) (\(R(L;{Z}_m)\)) denote the minimum integer \(N\) such that every function \(\Delta: \{1,2,\ldots,N\} \to \{0,1\}\) (\(A: \{1,2,\ldots,N\} \to {Z}_m\)) admits a solution of \(L\), say \((z_1,\ldots,z_m)\), such that \(\Delta(x_1) = \Delta(x_2) = \cdots = \Delta(x_m)\) (such that \(\sum_{i=1}^{m}\Delta(x_i) = 0\)). Define the system \(L_1(m)\) to consist of the inequality \(x_2 – x_1 \leq x_m – x_3\), and the system \(L_2(m)\) to consist of the inequality \(x_{m – 2}-x_{1} \leq x_m – x_{m-1}\); where \(x_1 < x_2 < \cdots < x_m\) in both \(L_1(m)\) and \(L_2(m)\). The main result of this paper is that \(R(L_1(m);2) = R(L_1(m);{Z}_m) = 2m\), and \(R(L_2(m);2) = 6m – 15\). Furthermore, we support the conjecture that \(R(L_1(m);2) = R(L_1(m);{Z}_m)\) by proving it for \(m = 5\).