Improved bounds for lines and $1$-separated sets in Euclidean Ramsey theory

Abstract

Let K be a $1$-separated set of diameter at most R−1, and let ℓ_m denote a collection of m points on a line, with consecutive points of distance $1$ apart. Conlon and Fox (2019) demonstrated a coloring of n-dimensional Euclidean space avoiding red congruent copies of ℓ₂ and blue congruent copies of K for |K| > 10000ⁿlog₂ R. We show here a stronger bound, that in fact |K| > (6.79 + o(1))ⁿlog R suffices for arbitrary $1$-separated K, while the improvement |K| > (5 + o(1))ⁿlog R holds in many cases, including when K = ℓ_m, or more generally when K is contained in a low-dimensional affine subspace. We also make a special study of the case when n=2, demonstrating a two-coloring of two-dimensional Euclidean space avoiding red copies of ℓ₂ and blue copies of ℓ₆₃₃₀. This latter result addresses a question of Erdős and Graham.

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BibTeX RIS

BibTeX

@misc{currier2026improved,
  title = {{Improved bounds for lines and $1$-separated sets in Euclidean Ramsey theory}},
  author = {Gabriel Currier and Param Mody and Zehan Xie and Jiaming Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2606.17194},
  eprint = {2606.17194},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.17194},
}

RIS

TY  - UNPB
TI  - Improved bounds for lines and $1$-separated sets in Euclidean Ramsey theory
AU  - Gabriel Currier
AU  - Param Mody
AU  - Zehan Xie
AU  - Jiaming Zhang
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2606.17194
UR  - https://arxiv.org/abs/2606.17194
AB  - Let K be a $1$-separated set of diameter at most R−1, and let ℓ_m denote a collection of m points on a line, with consecutive points of distance $1$ apart. Conlon and Fox (2019) demonstrated a coloring of n-dimensional Euclidean space avoiding red congruent copies of ℓ_2 and blue congruent copies of K for |K| > 10000^nlog_2 R. We show here a stronger bound, that in fact |K| > (6.79 + o(1))^nlog R suffices for arbitrary $1$-separated K, while the improvement |K| > (5 + o(1))^nlog R holds in many cases, including when K = ℓ_m, or more generally when K is contained in a low-dimensional affine subspace. We also make a special study of the case when n=2, demonstrating a two-coloring of two-dimensional Euclidean space avoiding red copies of ℓ_2 and blue copies of ℓ_6330. This latter result addresses a question of Erdős and Graham.
ER  -

Abstract

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