{
  "schema_version": 1,
  "problem_number": "OWR-1194-009",
  "title": "Maximal Gaps between PL and Linear Embedding Dimensions",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a finite simplicial complex K, let e_PL(K) and e_L(K) denote its minimum piecewise-linear and simplex-wise linear embedding dimensions. We prove that, for every d>=3, a finite d-complex satisfies e_PL(K)=d and e_L(K)=2d+1; this is the largest possible gap in that dimension. In dimension two we obtain the pair (3,5). These examples answer the minimum-dimension question recorded by Brehm and Sarkaria in the 2006 Oberwolfach report. The proof adapts Newman's Radon-pair, order-type and Janson method to make a random nonembedding obstruction survive every small set of facet deletions. Removing overlaps then produces a linear hypergraph, to which the PL realization lemma of Lee and Nevo applies. The construction is probabilistic and existential; no manifold hypothesis or explicit small facet list is asserted. This is an unrefereed, AI-assisted preprint.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.GT",
    "math.CO"
  ],
  "keywords": [
    "OWR-1194-009",
    "PL embedding",
    "linear embedding",
    "simplicial complex",
    "random hypergraph",
    "Janson inequality"
  ],
  "manuscript_version_date": "2026-10-25",
  "publication_date": "2026-10-25",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-25",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-1194-009/",
  "pdf_url": "https://eulersolve.org/papers/owr-1194-009/paper.pdf?v=2ffac5425caa",
  "doi": "10.5281/zenodo.23631404",
  "zenodo_record_url": "https://zenodo.org/records/23631404",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For every d>=3 there exists a finite d-dimensional simplicial complex with minimum PL embedding dimension d and minimum linear embedding dimension 2d+1, the maximum possible gap. A finite two-dimensional complex has the exact pair (3,5). This answers the full finite-complex existential question in OWR-1194-009, including dimensions two and three. The examples are probabilistic and need not be manifolds. The finite-complex existential question is answered in full. The probabilistic examples need not be manifolds and no explicit small facet list is claimed. The Radon-pair method of Newman and the PL realization lemma of Lee and Nevo are credited. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "2ffac5425caa480295cdeb9de9d082b0200115fefbdb1c4b7de419fae72162e6"
    },
    "source.zip": {
      "sha256": "ea40883842a349601529db1c0594acad785049e403a0e5629a6c3fd4d7897b72"
    },
    "verification_report.md": {
      "sha256": "3ff1b13b3b8ba5f9e3374db9138c711bda129eaa2bd26ae495bd11c16b663a01"
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  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}