PreprintCC BY 4.0Zenodo
Monochromatic symmetric subsets in finite colourings of infinite abelian groups
Abstract
We prove the assertion of Kourovka Problem 15.77 for every finite number of colours. If G is an infinite abelian group and α < |G|, every finite colouring of G has a monochromatic set S of cardinality exactly α such that 2g − S = S for some g ∈ G. For infinite α the argument uses a free-set lemma and a finite-colour directed-path lemma at α⁺; finite α follows from a ternary-cube embedding and the Hales–Jewett theorem.
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BibTeX
@misc{jayadevan2026monochromatic,
title = {{Monochromatic symmetric subsets in finite colourings of infinite abelian groups}},
author = {Achyuth Jayadevan},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23420558},
url = {https://zenodo.org/records/23420558},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Monochromatic symmetric subsets in finite colourings of infinite abelian groups
AU - Achyuth Jayadevan
PY - 2026
DA - 2026/10/16
PB - Zenodo
DO - 10.5281/zenodo.23420558
UR - https://zenodo.org/records/23420558
AB - We prove the assertion of Kourovka Problem 15.77 for every finite number of colours. If G is an infinite abelian group and α < |G|, every finite colouring of G has a monochromatic set S of cardinality exactly α such that 2g − S = S for some g ∈ G. For infinite α the argument uses a free-set lemma and a finite-colour directed-path lemma at α⁺; finite α follows from a ternary-cube embedding and the Hales–Jewett theorem.
ER -