{
  "abstract": "Define $\\nabla(n)$ as the least possible number of cells that must be removed from the $n\\times n$ square board in order to leave no cycles. Put $L_n=\\left\\lceil\\left((n-1)^2+1\\right)/3\\right\\rceil$. The results of Luccio, Madelaine-Stewart, and Lien-Fu-Shih imply that $\\nabla(n)=L_n$ in case $n\\equiv 1\\pmod 6$ and $n\\equiv 4\\pmod 6$, while for the remaining $4$ residue classes $\\nabla(n)=L_n$ or $\\nabla(n)=L_n+1$. In this paper, we provide optimal constructions in the odd residue classes $n\\equiv3,5\\pmod6$, (proving $\\nabla(n)=L_n$) and a geometric obstruction showing that equality in lower bound is impossible when $n\\equiv0,2\\pmod6$ (yielding $\\nabla(n)=L_n+1$). Starting with $n=3$, the resulting sequence is thus given by \\[ \\nabla(n)=\\bigg{\\lceil}\\frac{n^2}{3}-\\frac{n}{6}+\\frac{4}{3}\\bigg{\\rceil}-\\bigg{\\lceil}\\frac{n}{2}\\bigg{\\rceil}. \\]",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20034",
  "author_total": 1,
  "authors": [
    {
      "name": "Giedrius Alkauskas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b4mftavm4jppq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20034",
  "fulltext_url": null,
  "id": "b4mftavm4jppq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20034",
  "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|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7222349106",
  "pdf_url": "https://arxiv.org/pdf/2610.20034",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b4mftavm4jppq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.20034",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20034"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Maximal subsets of $n\\times n$ board without cycles",
  "title_text": "Maximal subsets of n× n board without cycles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/b4mftavm4jppq",
  "venue": "arXiv",
  "versions": []
}