{
  "abstract": "We study the largest size $C(n)$ of a subset of the $n\\times n$ integer grid containing no isosceles triple, including equally spaced collinear triples. We prove $C(n)=Ω\\left(n\\sqrt{\\log\\log n/\\log n}\\right)$ by applying the sparse hypergraph coloring theorem of Cooper and Mubayi. Elementary geometry and primitive-direction counts give maximum degree $O(n^2\\log n)$ and pair-codegree at most $5n$ for the forbidden-triple hypergraph. The resulting bound improves the explicit guarantee in PatternBoost by a factor of order $\\sqrt{\\log\\log n}$. Recovered interaction records document how a Codex literature proposal, human route selection, finite verification, and author-relayed review informed proof synthesis and repair. We connect these actions to successive proof artifacts and extract two case-derived checking practices: tracking objects, parameters and conclusions in theorem applications, and seeding a definitional omission to test a finite verifier. Separately implemented enumerators agree on complete edge sets for every $2\\le n\\le12$; suppressing the degenerate-case branch loses exactly the equally spaced collinear triples. These records document assistance under human direction, and finite checks support implementation consistency rather than the asymptotic theorem itself.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04194",
  "author_total": 1,
  "authors": [
    {
      "name": "Yukai Song"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7gf9gmm7ke68v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04194",
  "fulltext_url": null,
  "id": "7gf9gmm7ke68v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04194",
  "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"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7220724563",
  "pdf_url": "https://arxiv.org/pdf/2610.04194",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7gf9gmm7ke68v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.04194",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04194"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Sparse coloring for isosceles-free grids: an AI-assisted proof case study",
  "title_text": "Sparse coloring for isosceles-free grids: an AI-assisted proof case study",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/7gf9gmm7ke68v",
  "venue": "arXiv",
  "versions": []
}