{
  "abstract": "Simon's conjecture (1994) asserts that any pure $d$-dimensional shellable complex on $n$ vertices can be extended to the $d$-skeleton of the simplex on $n$ vertices, one facet at a time, while maintaining shellability. Bolognini and Sentinelli (2026) recently disproved it for every $d \\geq 3$. We show that the conjecture becomes true once $d-2$ new vertices are allowed, and that the same holds with $k$-decomposability in place of shellability for any $k \\geq 1$. We also show that the number $d-2$ cannot be lowered. Inflating the counterexample of Bolognini and Sentinelli gives, for every $d \\geq 3$, a $1$-decomposable complex that cannot be extended in this way with only $d-3$ new vertices, even while merely maintaining shellability.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32721",
  "author_total": 1,
  "authors": [
    {
      "name": "SuHo Oh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7h40e4gg242xc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32721",
  "fulltext_url": null,
  "id": "7h40e4gg242xc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32721",
  "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": "W7214806648",
  "pdf_url": "https://arxiv.org/pdf/2609.32721",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7h40e4gg242xc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2609.32721",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32721"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Completing shellings with $d-2$ extra vertices",
  "title_text": "Completing shellings with d−2 extra vertices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/7h40e4gg242xc",
  "venue": "arXiv",
  "versions": []
}