{
  "abstract": "We formulate a geometric version of the Erdős-Hajnal conjecture that applies to finite projective geometries rather than graphs, in both its usual 'induced' form and the multicoloured form. The multicoloured conjecture states, roughly, that a colouring $c$ of the points of $PG(n-1,q)$ containing no copy of a fixed colouring $c_0$ of $PG(k-1,q)$ for small $k$ must contain a subspace of dimension polynomial in $n$ that avoids some colour. If $(k,q) = (2,2)$, then $c_0$ is a colouring of a three-element 'triangle', and there are three essentially different cases, all of which we resolve. We derive both the cases where $c_0$ assigns the same colour to two different elements from a recent breakthrough result in additive combinatorics due to Kelley and Meka. We handle the case that $c_0$ is a 'rainbow' colouring by proving that rainbow-triangle-free colourings of projective geometries are exactly those that admit a certain decomposition into two-coloured pieces. This is closely analogous to a theorem of Gallai on rainbow-triangle-free coloured complete graphs. We also show that existing structure theorems resolve certain two-coloured cases where $(k,q) = (2,3)$, and $(k,q) = (3,2)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 6,
  "authors": [
    {
      "name": "Carolyn Chun"
    },
    {
      "name": "James Dylan Douthitt"
    },
    {
      "name": "Wayne Ge"
    },
    {
      "name": "Tony Huynh"
    },
    {
      "name": "Matthew E. Kroeker"
    },
    {
      "name": "Peter Nelson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/66cg0ht8ekqpz.bib",
  "cited_by": 0,
  "doi": "10.37236/14754",
  "fulltext_url": null,
  "id": "66cg0ht8ekqpz",
  "issns": [
    "1077-8926"
  ],
  "landing_url": "https://www.combinatorics.org/ojs/index.php/eljc/article/view/v33i4p5",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": "https://www.combinatorics.org/ojs/index.php/eljc/article/download/v33i4p5/pdf",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": "The Electronic Journal of Combinatorics",
  "ris": "https://research.draconitic.com/paper/66cg0ht8ekqpz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:24Z",
      "native_id": "10.37236/14754",
      "source": "crossref",
      "url": "https://www.combinatorics.org/ojs/index.php/eljc/article/view/v33i4p5"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "issn:1077-8926",
      "subfield": "combinatorics"
    }
  ],
  "title": "Rainbow Triangles and the Erdős-Hajnal Problem in Projective Geometries",
  "title_text": "Rainbow Triangles and the Erdős-Hajnal Problem in Projective Geometries",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:18:24Z",
  "url": "https://research.draconitic.com/paper/66cg0ht8ekqpz",
  "venue": "The Electronic Journal of Combinatorics",
  "versions": []
}