PreprintCC BY 4.0arXiv
Extensions of Hindman's theorem via finite colorings of topological groups
Abstract
We study the partition regular properties of topological groups, proving several extensions of Hindman theorem where monochromatic sets of finite sums are required to satisfy additional topological constraints. In particular, our results imply that for every nowhere dense set C ⊆ ℝn, there exists an open set P ⊇ C such that for any finite coloring of ℚn ∖ P, there is a family 𝒜 of sequences in ℚn ∖ P which satisfies the following properties: (i) for each A∈𝒜, the set FS(A) of finite sums of A is a closed discrete subset of ℝn; (ii) the set ⋃A∈𝒜FS(A) is monochromatic; and (iii) the set ⋃A∈𝒜FS(A) is dense in an open unbounded subset of ℝn.
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BibTeX
@misc{bardyla2026extensions,
title = {{Extensions of Hindman's theorem via finite colorings of topological groups}},
author = {Serhii Bardyla},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.31088},
eprint = {2608.31088},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.31088},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Extensions of Hindman's theorem via finite colorings of topological groups
AU - Serhii Bardyla
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.31088
UR - https://arxiv.org/abs/2608.31088
AB - We study the partition regular properties of topological groups, proving several extensions of Hindman theorem where monochromatic sets of finite sums are required to satisfy additional topological constraints. In particular, our results imply that for every nowhere dense set C ⊆ ℝ^n, there exists an open set P ⊇ C such that for any finite coloring of ℚ^n ∖ P, there is a family 𝒜 of sequences in ℚ^n ∖ P which satisfies the following properties: (i) for each A∈𝒜, the set FS(A) of finite sums of A is a closed discrete subset of ℝ^n; (ii) the set ⋃_A∈𝒜FS(A) is monochromatic; and (iii) the set ⋃_A∈𝒜FS(A) is dense in an open unbounded subset of ℝ^n.
ER -