{
  "abstract": "Kalai's cube--simplex conjecture asserts that for all positive integers $\\ell,k$, there is an integer $f(\\ell,k)$ such that every polytope of dimension at least $f(\\ell,k)$ has either a simplex $\\ell$-face or a cube $k$-face; let $f_s(\\ell,k)$ denote the threshold restricted to simple polytopes. Finiteness of $f(\\ell,k)$ is known only for $\\ell,k \\leq 2$. In addition, Kalai proved that $f_s(2,k) \\leq 2k^2$. Here we prove that $f_s(\\ell,k)$ is finite for all $\\ell \\geq 2$ and $k \\geq 3$, the first such result beyond $\\ell = 2$, with $f_s(2,k) \\leq 2k^2-1$ and $f_s(\\ell,k) \\leq \\tfrac{1}{2}k^2\\ell\\,2^k$ for $\\ell \\geq 3$. In the opposite direction, we obtain the lower bounds $f(\\ell,k) \\geq (5\\lfloor \\ell/2 \\rfloor + (\\ell \\bmod 2) - 1)(k-1)+1$ and $f_s(\\ell,k) \\geq \\max\\{4,\\,2(\\ell-1)\\}(k-1)+1$. A companion question asks for the minimum possible size of a 3-face within a higher-dimensional polytope. Meisinger, Kleinschmidt and Kalai proved that every rational $d$-polytope with $d \\geq 9$ has a $3$-face with fewer than $78$ vertices or fewer than $78$ facets. Here we improve their bound: every convex polytope of dimension at least $15$ has a $3$-face with at most $13$ facets. One step of our proof requires an explicit exact rational certificate or identity on flag numbers. This certificate is computed using linear programming.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.00397",
  "author_total": 5,
  "authors": [
    {
      "name": "Jesús A. De Loera"
    },
    {
      "name": "Ethan X. Fang"
    },
    {
      "name": "Shengtao Guo"
    },
    {
      "name": "Junwei Lu"
    },
    {
      "name": "Hailun Zheng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6rx80veg2wh2f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.00397",
  "fulltext_url": null,
  "id": "6rx80veg2wh2f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.00397",
  "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": "W7206149360",
  "pdf_url": "https://arxiv.org/pdf/2609.00397",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6rx80veg2wh2f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.00397",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.00397"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On Unavoidable Faces of High-Dimensional Polytopes",
  "title_text": "On Unavoidable Faces of High-Dimensional Polytopes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/6rx80veg2wh2f",
  "venue": "arXiv",
  "versions": []
}