A general lower bound on the weak Schur number

Q2 Mathematics
L. Boza, M.P. Revuelta, M.I. Sanz
{"title":"A general lower bound on the weak Schur number","authors":"L. Boza,&nbsp;M.P. Revuelta,&nbsp;M.I. Sanz","doi":"10.1016/j.endm.2018.06.024","DOIUrl":null,"url":null,"abstract":"<div><p>For integers <em>k</em>, <em>n</em> with <span><math><mi>k</mi><mo>,</mo><mi>n</mi><mo>≥</mo><mn>1</mn></math></span>, the <em>n</em>-<em>color weak Schur number</em> <span><math><mi>W</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> is defined as the least integer <em>N</em>, such that for every <em>n</em>-coloring of the integer interval [1, <em>N</em>], there exists a monochromatic solution <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> in that interval to the equation <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mo>…</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>, with <span><math><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≠</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span>, when <span><math><mi>i</mi><mo>≠</mo><mi>j</mi></math></span>. We show a relationship between <span><math><mi>W</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span> and <span><math><mi>W</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and a general lower bound on the <span><math><mi>W</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> is obtained.</p></div>","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2018-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2018.06.024","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Electronic Notes in Discrete Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S157106531830115X","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 4

Abstract

For integers k, n with k,n1, the n-color weak Schur number WSk(n) is defined as the least integer N, such that for every n-coloring of the integer interval [1, N], there exists a monochromatic solution x1,,xk,xk+1 in that interval to the equation x1+x2++xk=xk+1, with xixj, when ij. We show a relationship between WSk(n+1) and WSk(n) and a general lower bound on the WSk(n) is obtained.

弱舒尔数的一般下界
对于k,n, k,n≥1的整数k,n色弱舒尔数WSk(n)定义为最小的整数n,使得对于整数区间[1,n]的每一个n色,在该区间存在方程x1+x2+…+xk=xk+1的单色解x1,…,xk,xk+1,且当i≠j时,x1+x2+…+xk=xk+1。我们证明了WSk(n+1)和WSk(n)之间的关系,并得到了WSk(n)的一般下界。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
Electronic Notes in Discrete Mathematics
Electronic Notes in Discrete Mathematics Mathematics-Discrete Mathematics and Combinatorics
CiteScore
1.30
自引率
0.00%
发文量
0
期刊介绍: Electronic Notes in Discrete Mathematics is a venue for the rapid electronic publication of the proceedings of conferences, of lecture notes, monographs and other similar material for which quick publication is appropriate. Organizers of conferences whose proceedings appear in Electronic Notes in Discrete Mathematics, and authors of other material appearing as a volume in the series are allowed to make hard copies of the relevant volume for limited distribution. For example, conference proceedings may be distributed to participants at the meeting, and lecture notes can be distributed to those taking a course based on the material in the volume.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信