{
  "abstract": "Fix one colour class in the checkerboard colouring of an $n\\times n$ integer grid, and let $M_4(n,\\varepsilon)$ be the largest subset having at most two points in every row, column, and diagonal of slopes $\\pm1$. We prove the near-saturation bound $M_4(n,\\varepsilon)\\leq2n-4$ for $n\\geq6$. The proof uses first and second moments of the four line families: a hypothetical set of size $2n-3$ produces row, column, and diagonal deficits whose exact moment identities contradict Cauchy--Schwarz. A finite argument handles $n=6$. The same identity extends to arbitrary deficit multisets. It gives $M_4(n,\\varepsilon)\\leq2n-d$ whenever $d\\geq4$ is an integer and $n\\geq3d-4$, and an entirely discrete asymptotic estimate \\[ M_4(n,\\varepsilon)\\leq(\\sqrt{21}-3)n+8. \\] We also prove a general continuum-to-discrete theorem for the associated four-direction fractional packing problem. Applying it to the exact continuum dual certificate constructed in earlier work yields, for both colours, \\[ L_{\\mathrm{mono}}(n,\\varepsilon)\\leqαn+O(1), \\qquad α\\approx1.5768233968738, \\] and hence the same upper bound for $M_4$ and for checkerboard no-three-in-line sets.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12340",
  "author_total": 3,
  "authors": [
    {
      "name": "Jujhar Aujla"
    },
    {
      "name": "Thomas Prellberg"
    },
    {
      "name": "Nirvair Sandhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0yg8zpz0dh5ta.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12340",
  "fulltext_url": null,
  "id": "0yg8zpz0dh5ta",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12340",
  "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"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7213117552",
  "pdf_url": "https://arxiv.org/pdf/2609.12340",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0yg8zpz0dh5ta.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.12340",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12340"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Moment obstructions and continuum-to-discrete bounds for checkerboard no-three-in-line sets",
  "title_text": "Moment obstructions and continuum-to-discrete bounds for checkerboard no-three-in-line sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/0yg8zpz0dh5ta",
  "venue": "arXiv",
  "versions": []
}