{
  "abstract": "We study the Shield number of the stable marriage problem, defined as the minimum, over all preference profiles of size n, of the maximum number of blocking pairs attainable by a complete matching. We establish the universal quadratic lower bound $σ(n) \\ge \\lceil n(n-2)/6 \\rceil$ using a rearrangement inequality and show that this bound is the strongest consequence obtainable from uniform first-moment arguments. We further prove a sharp min-rearrangement inequality and construct an explicit cyclic Hollow-Shell instance attaining $\\lfloor (n-1)^2/4 \\rfloor$ blocking pairs. We also establish sufficient Hall-type conditions for biclique realizability, prove structural limitations of uniform averaging and two-parameter linear assignment objectives, and provide complete proofs for all unconditional results. Computer-assisted verification is used only for explicitly identified computational statements and is clearly distinguished from mathematical proofs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17418",
  "author_total": 1,
  "authors": [
    {
      "name": "Yoshiteru Ishida"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aj19qnb3th7xt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17418",
  "fulltext_url": null,
  "id": "aj19qnb3th7xt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17418",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2609.17418",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aj19qnb3th7xt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.17418",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17418"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Quadratic Lower Bound for the Shield Number of the Stable Marriage Problem: Rearrangement, Extremal Construction, and Biclique Realizability",
  "title_text": "A Quadratic Lower Bound for the Shield Number of the Stable Marriage Problem: Rearrangement, Extremal Construction, and Biclique Realizability",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/aj19qnb3th7xt",
  "venue": "arXiv",
  "versions": []
}