On a Zero-Sum Generalization of a Variation of Schur’s Equation

摘要

Let m≥3 be a positive integer, and let Z_m denote the cyclic group of residues modulo m. Furthermore, let R(L_m;2)(R(L)M;Z_m)) denote the minimum integer N such that for every function △:{1,2,...,N}→{0,1}(△:{1,2,..,N}→Z_m) there exist m integers x_1＜x_2＜…＜x_m satisfying ∑~(m-1)_(i=1)x_i＜x_m and △(x_1)=△(x_2)= …=△(x_m) (and ∑~m_(i=1)△(x_i)=0)). It is shown that R(L_m;2)=R(L_m;Z_m). for every odd prime m.