Brown-Gerver-Ramsey Theorems in Small Dimensions

2026-09-17

AI Summary

研究解决了在特定维度中避免一定数量共线点的无限行走问题，通过构造方法改进了之前的结果，并推广到弱阿贝尔幂自由词的存在性。

Abstract

We consider infinite walks in $\mathbb{N}^k$ with standard unit basis vector steps that avoid $t$ collinear points, and show that these walks exist for $(k,t) \in \{(6,3), (4,4), (3,7)\}$. In particular, our construction for $k = 3$ improves the previous bound $189$, obtained by Lidbetter, to $7$. Our results also imply the existence of infinite words over small finite alphabets that are weakly abelian squarefree (resp., weakly abelian cubefree, weakly abelian 6th-power-free).

Problem

• infinite walks
• collinear points
• small dimensions
• weakly abelian power-free

Innovation

• infinite walks
• avoid t collinear points
• small dimensions
• weakly abelian power-free