On a Zero-Sum Generalization of a Variation of Schur¡¯s Equation

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• 作者：Arie Bialostocki ; Rasheed Sabar and Daniel Schaal
• 关键词：zero ; sum ; Erdő ; s ; Ginzburg ; Ziv ; Ramsey
• 刊名：Graphs and Combinatorics
• 出版年：2008
• 出版时间：November, 2008
• 年：2008
• 卷：24
• 期：6
• 页码：511-518
• 全文大小：134.0 KB

文摘

Let m\geqslant 3m\geqslant 3 be a positive integer, and let \mathbbZ_m\mathbb{Z}_m denote the cyclic group of residues modulo m. Furthermore, let R(L_m; 2)(R(L_m;\mathbbZ_m))R(L_m; 2)(R(L_m;\mathbb{Z}_m)) denote the minimum integer N such that for every function D: {1,2,?,N}? {0,1} (D: {1,2,?,N}? \mathbbZ_m)\Delta: \{1,2,\ldots,N\}\rightarrow \{0,1\}\,(\Delta: \{1,2,\ldots,N\}\rightarrow \mathbb{Z}_m) there exist m integers x₁ < x₂ < ? < x_mx_1 satisfying ?_i=1m-1x_i < x_m\sum_{i=1}^{m-1}x_i and D(x₁)=D(x₂)=? = D(x_m)\Delta(x_1)=\Delta(x_2)=\cdots=\Delta(x_m) (and ?_i=1mD(x_i)=0\sum_{i=1}^{m}\Delta(x_i)=0). It is shown that R(L_m;2)=R(L_m;\mathbbZ_m)R(L_{m};2)=R(L_m;\mathbb{Z}_m) for every odd prime m.