Upper and Lower Bounds on Zero-Sum Generalized Schur Numbers

Let \(S_{\mathfrak{z}}(k,r)\) be the least positive integer such that for any \(r\)-coloring \(\chi : \{1,2,\dots,S_{\mathfrak{z}}(k,r)\} \longrightarrow \{1, 2, \dots, r\}\), there is a sequence \(x_1, x_2, \dots, x_k\) such that \(\sum_{i=1}^{k-1} x_i = x_k\), and \(\sum_{i=1}^{k} \chi(x_i) \equiv...

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Bibliographic Details
Published in:arXiv.org 2018-08
Main Author: Metz, Erik
Format: Article
Language:English
Subjects:
Coloring
Lower bounds
Numbers
Online Access:Get full text
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Summary:Let \(S_{\mathfrak{z}}(k,r)\) be the least positive integer such that for any \(r\)-coloring \(\chi : \{1,2,\dots,S_{\mathfrak{z}}(k,r)\} \longrightarrow \{1, 2, \dots, r\}\), there is a sequence \(x_1, x_2, \dots, x_k\) such that \(\sum_{i=1}^{k-1} x_i = x_k\), and \(\sum_{i=1}^{k} \chi(x_i) \equiv 0 \pmod{r}\). We show that when \(k\) is greater than \(r\), \(kr - r - 1 \le S_{\mathfrak{z}}(k,r) \le kr - 1\), and when \(r\) is an odd prime, \(S_{\mathfrak{z}}(k,r)\) is in fact equal to \(kr - r\).
ISSN:2331-8422