{
  "abstract": "We prove a dimensional strengthening of the colorful Helly theorem. Let $\\mathcal{C}_1,\\dots,\\mathcal{C}_{d+1}$ be finite nonempty families of convex sets in $\\mathbb{R}^d$. If $C_1\\cap\\cdots\\cap C_{d+1}\\neq\\varnothing$ for every choice of $C_i\\in\\mathcal{C}_i$, then $\\sum_{i=1}^{d+1}\\dim(\\bigcap\\mathcal{C}_i)\\geq0$, where $\\dim\\varnothing=-1$. We also obtain a dimensional strengthening of a theorem of Kim and Lew, in which the intersections are taken over unions of color classes. For simplicial complexes, we introduce the link-Leray dimension, defined as the Leray number of the link for a face and as $-1$ for a nonface. Using this invariant, we prove a matroidal extension of the dimensional colorful Helly theorem for $d$-Leray complexes. We further establish a corresponding strengthening of the topological Kim--Lew theorem in two ranges of parameters. As a special case, we recover the topological colorful Helly theorem of Kalai and Meshulam. Finally, we construct counterexamples to the unrestricted topological extension, even under a stronger local condition.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.36242",
  "author_total": 1,
  "authors": [
    {
      "name": "Wei Rao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6c7yrky1pbjf8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.36242",
  "fulltext_url": null,
  "id": "6c7yrky1pbjf8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.36242",
  "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": "W7214957935",
  "pdf_url": "https://arxiv.org/pdf/2609.36242",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6c7yrky1pbjf8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.36242",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.36242"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Dimensional colorful Helly theorems and topological variants",
  "title_text": "Dimensional colorful Helly theorems and topological variants",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/6c7yrky1pbjf8",
  "venue": "arXiv",
  "versions": []
}