PreprintCC BY 4.0arXiv
Brown-Gerver-Ramsey Theorems in Small Dimensions
Abstract
We consider infinite walks in ℕk with standard unit basis vector steps that avoid t collinear points, and show that these walks exist for (k,t) ∈ {(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).
Related
BibTeX
@misc{cambie2026browngerverramsey,
title = {{Brown-Gerver-Ramsey Theorems in Small Dimensions}},
author = {Stijn Cambie and Erik Kalviainen and J. Shallit},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20366},
eprint = {2609.20366},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20366},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Brown-Gerver-Ramsey Theorems in Small Dimensions
AU - Stijn Cambie
AU - Erik Kalviainen
AU - J. Shallit
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20366
UR - https://arxiv.org/abs/2609.20366
AB - We consider infinite walks in ℕ^k with standard unit basis vector steps that avoid t collinear points, and show that these walks exist for (k,t) ∈ {(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).
ER -