{
  "abstract": "For a graph $H$, let $\\operatorname{ex}(Q_n, H)$ be the largest number of edges in a subgraph of the hypercube $Q_n$ of dimension $n$ that contains no subgraph isomorphic to $H$. We define $π_\\square(H)=\\lim_{n\\to\\infty}\\operatorname{ex}(Q_n,H)/|E(Q_n)|$ as the Turán density of $H$ in the hypercube. Although determining $π_\\square(H)$ for general $H$ remains widely open, Conlon found a large class of graphs with zero Turán density in a hypercube. In this note, we address the case when $π_{\\square} (H)>0$. If a graph $H$ cannot be embedded in an edge layer of a hypercube, then $π_{\\square} (H)\\geq 1/2$, as can be seen by taking every other edge layer of $Q_n$. Previously known examples of layered graphs of positive Turán density in a hypercube include graphs containing cycles of length $6$ or $10$, as well as examples of girth $8$. Also, to the best of our knowledge, all previously known examples of graphs with Turán densities arbitrarily close to one are not layered. We show that for every $g \\geq 3$, there are layered graphs of girth greater than $g$ whose Turán densities in a hypercube are arbitrarily close to one. Moreover, these graphs have an induced embedding into an edge layer of a hypercube.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30544",
  "author_total": 2,
  "authors": [
    {
      "name": "Maria Axenovich"
    },
    {
      "name": "Marko Pejić"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f08qa0236sj5t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30544",
  "fulltext_url": null,
  "id": "f08qa0236sj5t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30544",
  "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",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7204947654",
  "pdf_url": "https://arxiv.org/pdf/2608.30544",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f08qa0236sj5t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2608.30544",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30544"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On high-girth layered graphs of large Turán density in a hypercube",
  "title_text": "On high-girth layered graphs of large Turán density in a hypercube",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/f08qa0236sj5t",
  "venue": "arXiv",
  "versions": []
}