{
  "abstract": "The classical no-three-in-line problem asks for the largest number $D(n)$ of points that can be chosen from an $n\\times n$ grid with no three collinear; it remains open whether the elementary upper bound $2n$ is always attainable. We study a checkerboard-restricted variant in which all chosen points are monochromatic under the colouring of the grid by the parity of $x+y$. If $D_{\\mathrm{mono}}(n)$ denotes the largest number of monochromatic points with no three collinear, the monochromatic diagonals already give $D_{\\mathrm{mono}}(n)\\le 2n-2$. The main object of the paper is a four-direction linear-programming relaxation on a fixed colour class, using rows, columns and the two diagonal families of slopes $\\pm1$. For the ordinary square-grid problem this relaxation gives the trivial bound; on the checkerboard it is substantially tighter, and for $2\\le n\\le 16$ its floor agrees with the exact single-colour optimum except at four side lengths, where the gap is one. After symmetry reduction the dual relaxation has three one-dimensional reduced forms, according to the parity of $n$ and the chosen colour class. The central construction is an exact continuum dual certificate for the continuum problem associated with the scaled symmetry-reduced odd-fat case: explicit nonnegative functions $A$ and $B$ satisfying the continuum obstacle inequalities, with objective value the middle real root $α\\approx1.5768$ of $401α^3-1744α^2+2240α-768=0$. Combined with a continuum-to-discrete sampling theorem of Aujla, Prellberg and Sandhu, this certificate yields the asymptotic upper bound $D_{\\mathrm{mono}}(n)\\leαn+O(1)$. Finite LP computations are consistent with $α$ as the exact limiting slope of the relaxation, and the exact small-$n$ data suggest, more speculatively, that the true checkerboard optimum tracks the same scale.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.09215",
  "author_total": 1,
  "authors": [
    {
      "name": "Thomas Prellberg"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6deasdvbw22mx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.09215",
  "fulltext_url": null,
  "id": "6deasdvbw22mx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.09215",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7160949440",
  "pdf_url": "https://arxiv.org/pdf/2605.09215",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6deasdvbw22mx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2605.09215",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.09215"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "No-three-in-line sets on the checkerboard grid",
  "title_text": "No-three-in-line sets on the checkerboard grid",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/6deasdvbw22mx",
  "venue": "arXiv",
  "versions": []
}