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

Abstract

Let $m\ge 4$ 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 $\mathrm{\Delta }:\{1,2,\dots ,N\}\to \{0,1\}$ ($A:\{1,2,\dots ,N\}\to Z_m$) admits a solution of $L$, say $(z_1,\dots ,z_m)$, such that $\mathrm{\Delta }(x_1)=\mathrm{\Delta }(x_2)=\cdots =\mathrm{\Delta }(x_m)$ (such that $\sum _{i=1}^m\mathrm{\Delta }(x_i)=0$). Define the system $L_1(m)$ to consist of the inequality $x_2–x_1\le x_m–x_3$, and the system $L_2(m)$ to consist of the inequality $x_{m–2}-x_1\le x_m–x_{m-1}$; 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 $m=5$.