{
  "abstract": "A hypergraph is linear if every pair of vertices is contained in at most one hyperedge. We study linear Turán numbers of acyclic $r$-uniform hypergraphs, extending earlier work of Gyárfás, Ruszinkó, and Sárközy from uniformity $3$ to general $r$, with emphasis on hypertrees with four hyperedges. For the linear path $P_4^r$, we prove $\\operatorname{ex}^{\\mathrm{lin}}_r(n,P_4^r)\\le (r+1)n/r$, with equality precisely for disjoint unions of Steiner systems $S(2,r,r^2)$, whenever such systems exist. We also give counterexamples to a structural claim used in a previously proposed proof for $r=4$. The same bound and equality characterization hold for the broom $B_4^r$. For the crown $E_4^r$, writing $B=\\{v:d(v)\\le r\\}$ and $C=\\{v:r+1\\le d(v)\\le2r-1\\}$, we prove $|E(H)|\\le r|B|/(r-1)+(2r-1)|C|/r$, and hence $\\operatorname{ex}^{\\mathrm{lin}}_r(n,E_4^r)\\le(2r-1)n/r$. We also give a lower-bound construction and formulate a sharper conjecture. Finally, for the linear star $S_k^r$, we prove $\\operatorname{ex}^{\\mathrm{lin}}_r(n,S_k^r)\\le n(k-1)/r$, with equality precisely for $(k-1)$-regular linear $r$-uniform hypergraphs, whenever such hypergraphs exist. Under suitable divisibility and existence assumptions, we also construct $T_k^r$-free examples with $n(k-1)/r$ edges for every linear $r$-uniform hypertree $T_k^r$ with $k$ hyperedges.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.16854",
  "author_total": 2,
  "authors": [
    {
      "name": "Rajat Adak"
    },
    {
      "name": "Pragya Verma"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bzzccx0b45qmm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.16854",
  "fulltext_url": null,
  "id": "bzzccx0b45qmm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.16854",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7169872914",
  "pdf_url": "https://arxiv.org/pdf/2607.16854",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bzzccx0b45qmm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2607.16854",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.16854"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Linear Turán Numbers of Uniform Hypertrees",
  "title_text": "Linear Turán Numbers of Uniform Hypertrees",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/bzzccx0b45qmm",
  "venue": "arXiv",
  "versions": []
}