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arXiv · 2609.20366

Brown-Gerver-Ramsey Theorems in Small Dimensions

Stijn Cambie·Erik Kalviainen·J. Shallit
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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).

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Stijn Cambie, Erik Kalviainen, J. Shallit. 2026-09-17. Brown-Gerver-Ramsey Theorems in Small Dimensions. https://arxiv.org/abs/2609.20366

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