{
  "schema_version": 1,
  "problem_number": "OWR-1703876-006",
  "title": "A Proof of Paták's kb+1 Conjecture for Constrained Stars, with Improved Bounds for Complete Graphs",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "In an Oberwolfach report from 2020, Paták considered closure operators on topological spaces whose closures have at most b path-connected components. He noted that C(b+1,2)(k−1)+b+1 points suffice for a constrained drawing of the star K_{1,k}, conjectured that kb+1 points suffice, and asked two further questions: whether the bounds for complete graphs K_n can be improved, and whether the method extends to higher homology or homotopy. In the journal version (J. Graph Theory, 2025) the conjecture is stated for b-iatlon graphs and proved there for b ≤ 2. We prove the conjecture for all k and b. Every b-iatlon graph with kb+1 vertices contains a constrained copy of K_{1,k}. For every closure operator as above, every set of kb+1 points admits a constrained drawing of K_{1,k}. The bound kb+1 is sharp in both settings. The proof rests on a colouring lemma for families of graphs indexed by the subsets of a finite set and growing with the subset; it extends the bound χ ≤ Δ+1. As a consequence, 1+b+⋯+b^{n−1} points force a constrained copy of K_n, improving the previous bound O(b^{2n−3}). Combined with Paták's topological arguments, this lowers his bounds on Radon and Helly numbers in R^d from O(b^{2d+3}) to O(b^{d+2}), and his bound on Radon numbers on a fixed closed surface from O(b^6) to O(b^3). In the b-iatlon setting, Ramsey numbers give lower bounds, which show that the exponent n−1 is optimal for n = 3, 4. Exact values for complete graphs remain open, and the question on higher homology and homotopy is not addressed. This is an unrefereed note.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.CO",
    "math.MG"
  ],
  "keywords": [
    "constrained drawings",
    "b-iatlon graphs",
    "closure operators",
    "Radon number",
    "Helly number",
    "Ramsey numbers",
    "graph colouring",
    "Oberwolfach Reports",
    "OWR-1703876-006",
    "math.CO",
    "math.MG",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-09-30",
  "publication_date": "2026-09-30",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-30",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-1703876-006/",
  "pdf_url": "https://eulersolve.org/papers/owr-1703876-006/paper.pdf?v=b2db3bc623ae",
  "doi": "10.5281/zenodo.23062843",
  "zenodo_record_url": "https://zenodo.org/records/23062843",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Solves the first of Paták's three questions (OWR 30/2020): kb+1 points force a constrained drawing of the star K_{1,k}, and this bound is sharp. The bounds for complete graphs are improved but exact values remain open, and the question on higher homology or homotopy is not addressed.",
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    "source.zip": {
      "sha256": "569b43d1e17495ddb56a1b35777995ef9c231f081e7952c9512737447de5bbbd"
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    "verification_report.md": {
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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."
}