Gerver-Ramsey共线性问题的改进界

SIAM J. Discret. Math. Pub Date : 2023-03-25 DOI:10.48550/arXiv.2303.14579

T. Lidbetter

{"title":"Gerver-Ramsey共线性问题的改进界","authors":"T. Lidbetter","doi":"10.48550/arXiv.2303.14579","DOIUrl":null,"url":null,"abstract":"Let $S$ be a finite subset of $\\mathbb{Z}^n$. A vector sequence $(\\mathbf{z}_i)$ is an $S$-walk if and only if $\\mathbf{z}_{i+1} - \\mathbf{z}_i$ is an element of $S$ for all $i$. Gerver and Ramsey showed in 1979 that for $S\\subset \\mathbb{Z}^3$ there exists an infinite $S$-walk in which no $5^{11} + 1=48{\\small,}828{\\small,}126$ points are collinear. Here, we use the same general approach, but with the aid of a computer search, to improve the bound to $189$.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":"72 1","pages":"113718"},"PeriodicalIF":0.0000,"publicationDate":"2023-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Improved Bound for the Gerver-Ramsey Collinearity Problem\",\"authors\":\"T. Lidbetter\",\"doi\":\"10.48550/arXiv.2303.14579\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $S$ be a finite subset of $\\\\mathbb{Z}^n$. A vector sequence $(\\\\mathbf{z}_i)$ is an $S$-walk if and only if $\\\\mathbf{z}_{i+1} - \\\\mathbf{z}_i$ is an element of $S$ for all $i$. Gerver and Ramsey showed in 1979 that for $S\\\\subset \\\\mathbb{Z}^3$ there exists an infinite $S$-walk in which no $5^{11} + 1=48{\\\\small,}828{\\\\small,}126$ points are collinear. Here, we use the same general approach, but with the aid of a computer search, to improve the bound to $189$.\",\"PeriodicalId\":21749,\"journal\":{\"name\":\"SIAM J. Discret. Math.\",\"volume\":\"72 1\",\"pages\":\"113718\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-03-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"SIAM J. Discret. Math.\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.48550/arXiv.2303.14579\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM J. Discret. Math.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.48550/arXiv.2303.14579","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 0

摘要

设$S$是$\mathbb{Z}^n$的有限子集。向量序列$(\mathbf{z}_i)$是$S$-walk当且仅当$\mathbf{z}_{i+1} - \mathbf{z}_i$对于所有$i$都是$S$的元素。Gerver和Ramsey在1979年证明了对于$S\子集\mathbb{Z}^3$存在一个无限$S$-walk，其中没有$5^{11}+ 1=48{\small，}828{\small，}126$点共线。在这里，我们使用相同的一般方法，但借助计算机搜索，将绑定提高到$189$。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Improved Bound for the Gerver-Ramsey Collinearity Problem

Let $S$ be a finite subset of $\mathbb{Z}^n$. A vector sequence $(\mathbf{z}_i)$ is an $S$-walk if and only if $\mathbf{z}_{i+1} - \mathbf{z}_i$ is an element of $S$ for all $i$. Gerver and Ramsey showed in 1979 that for $S\subset \mathbb{Z}^3$ there exists an infinite $S$-walk in which no $5^{11} + 1=48{\small,}828{\small,}126$ points are collinear. Here, we use the same general approach, but with the aid of a computer search, to improve the bound to $189$.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

SIAM J. Discret. Math.

自引率

0.00%

发文量