{
  "abstract": "A classical theorem of Kaiser and Rabinovich implies that if every subfamily of size at most $\\lfloor\\log_2(d+1)\\rfloor+1$ of a finite family of compact connected sets in $\\mathbb{R}^d$ has a common axis-parallel hyperplane transversal, then the entire family has an orthogonal star transversal, i.e., a union of $d$ hyperplanes, one perpendicular to each coordinate axis. We study colorful variants of this result. First, we use standard KKM machinery to establish a colorful analogue with $d+1$ color classes and a $\\lfloor\\log_2(d+1)\\rfloor+1$ transversal hypothesis for rainbow subfamiles. Then we focus on reducing the required number of colors. To this end, we introduce the rainbow pushability $π_r(P)$ of a polytope and we use it to state and prove few-color variants of the colorful Carathéodory theorem and the colorful polytopal KKMS theorem. Applying the latter to the $d$-dimensional cube, we show that $\\lfloor\\log_2 d\\rfloor+2$ color classes suffice for our transversal problem, provided that every rainbow choice has a common axis-parallel hyperplane transversal. We also extend both colorful results to the setting of convex $(n,D)$-bodies and we show that three colors are needed for $d\\geq 2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17157",
  "author_total": 2,
  "authors": [
    {
      "name": "Leonardo Martínez-Sandoval"
    },
    {
      "name": "Ariadna Olvera-Sampieri"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/54w1rrg3qt16a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17157",
  "fulltext_url": null,
  "id": "54w1rrg3qt16a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17157",
  "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": "W7221748001",
  "pdf_url": "https://arxiv.org/pdf/2610.17157",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/54w1rrg3qt16a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.17157",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17157"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Pushed colorful polytopal KKMS Theorem and orthogonal star transversals",
  "title_text": "Pushed colorful polytopal KKMS Theorem and orthogonal star transversals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/54w1rrg3qt16a",
  "venue": "arXiv",
  "versions": []
}