{
  "abstract": "An $n$-colouring of the real line is a Lebesgue measurable function $f \\colon \\mathbb{R} \\to A$, where $A$ is a set of $n$ colours. An abelian square for $f$ is a finite interval $U\\subseteq \\mathbb{R}$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28376",
  "author_total": 3,
  "authors": [
    {
      "name": "Ville Salo"
    },
    {
      "name": "Ingrid Vukusic"
    },
    {
      "name": "Charlie Wilson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2jxxr298jb68m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28376",
  "fulltext_url": null,
  "id": "2jxxr298jb68m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28376",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7223109126",
  "pdf_url": "https://arxiv.org/pdf/2610.28376",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2jxxr298jb68m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.28376",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28376"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Avoiding abelian squares with three colours in the continuous setting",
  "title_text": "Avoiding abelian squares with three colours in the continuous setting",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/2jxxr298jb68m",
  "venue": "arXiv",
  "versions": []
}