Avoiding abelian squares with three colours in the continuous setting
Abstract
An n-colouring of the real line is a Lebesgue measurable function f : ℝ → A, where A is a set of n colours. An abelian square for f is a finite interval U⊆ ℝ where every colour shows up in the left and in the right half of U with exactly the same Lebesgue measure. It is known that every $2$-colouring contains an abelian square. On the other hand, Alon, Grytczuk, Laso{ń}, and Michałek (2009) proved that there exist $4$-colourings that avoid abelian squares, while the question for $3$-colourings remained open. We construct a $3$-colouring that avoids abelian squares, disproving a conjecture of Laso{ń} (2015). However, in contrast to the case of 4-colourings, the set of 3-colourings that avoid abelian squares is meagre. We show this using an argument based on the KKM lemma.
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BibTeX
@misc{salo2026avoiding,
title = {{Avoiding abelian squares with three colours in the continuous setting}},
author = {Ville Salo and Ingrid Vukusic and Charlie Wilson},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28376},
eprint = {2610.28376},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28376},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Avoiding abelian squares with three colours in the continuous setting
AU - Ville Salo
AU - Ingrid Vukusic
AU - Charlie Wilson
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28376
UR - https://arxiv.org/abs/2610.28376
AB - An n-colouring of the real line is a Lebesgue measurable function f : ℝ → A, where A is a set of n colours. An abelian square for f is a finite interval U⊆ ℝ where every colour shows up in the left and in the right half of U with exactly the same Lebesgue measure. It is known that every $2$-colouring contains an abelian square. On the other hand, Alon, Grytczuk, Laso{ń}, and Michałek (2009) proved that there exist $4$-colourings that avoid abelian squares, while the question for $3$-colourings remained open. We construct a $3$-colouring that avoids abelian squares, disproving a conjecture of Laso{ń} (2015). However, in contrast to the case of 4-colourings, the set of 3-colourings that avoid abelian squares is meagre. We show this using an argument based on the KKM lemma.
ER -