{
  "abstract": "Let $P_4^r$ denote the $r$-uniform expansion of the graph path with four edges. We prove that every $n$-vertex linear $r$-uniform $P_4^r$-free hypergraph has at most $(r+1)n/r$ edges, resolving a conjecture of Adak and Verma for every $r \\geq 2$. Equality holds precisely for vertex-disjoint unions of Steiner systems $S(2,r,r^2)$. The main ingredient is a sharp incidence-rank inequality. If $N(H)$ is the edge-vertex incidence matrix of a linear $r$-uniform hypergraph whose line graph is a cograph, then $(r+1)\\operatorname{rank}_{\\mathbb{R}} N(H) \\geq r|E(H)|$. Equality holds exactly when every edge-containing component is an $S(2,r,r^2)$. The proof follows the union-join decomposition of cographs. At a join node, the row-difference spaces of the co-components are mutually orthogonal, and the possible rank defect is determined by balanced co-components. Perron-Frobenius theory identifies the smallest balanced pieces as parallel classes of $r$ disjoint $r$-sets, while an orthogonality argument bounds their number by $r+1$. The equality case then reconstructs the Steiner system. The linear Turán bound follows from $\\operatorname{rank}_{\\mathbb{R}} N(H) \\leq |V(H)|$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20173",
  "author_total": 1,
  "authors": [
    {
      "name": "Mahesh Ramani"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1z60t6yfm5crf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20173",
  "fulltext_url": null,
  "id": "1z60t6yfm5crf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20173",
  "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|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7213589862",
  "pdf_url": "https://arxiv.org/pdf/2609.20173",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1z60t6yfm5crf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20173",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20173"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Linear Tur'an Numbers of Four-Edge Uniform Paths via Incidence Rank",
  "title_text": "Linear Tur'an Numbers of Four-Edge Uniform Paths via Incidence Rank",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/1z60t6yfm5crf",
  "venue": "arXiv",
  "versions": []
}